mmtheorems46 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4501-4600 - Page 46 of 108

Type	Label	Description
Statement
Theorem	xpmapen 4501	Equinumerosity law for set exponentiation of a cross product. Exercise 4.47 of [Mendelson] p. 255.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- ((A X. B) ^m C) ~~ ((A ^m C) X. (B ^m C))
Theorem	mapunen 4502	Equinumerosity law for set exponentiation of a disjoint union. Exercise 4.45 of [Mendelson] p. 255.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- ((A i^i B) = (/) -> (C ^m (A u. B)) ~~ ((C ^m A) X. (C ^m B)))
Theorem	pwen 4503	If two sets are equinumerous, then their power sets are equinumerous. Proposition 10.15 of [TakeutiZaring] p. 87.
|- B e. V   =>   |- (A ~~ B -> P~A ~~ P~B)
Theorem	ssenen 4504	Equinumerosity of equinumerous subsets of a set.
|- A e. V   &   |- B e. V   =>   |- (A ~~ B -> {x | (x (_ A /\ x ~~ C)} ~~ {x | (x (_ B /\ x ~~ C)})
Theorem	limenpsi 4505	A limit ordinal is equinumerous to a proper subset of itself.
|- Lim A   =>   |- (A e. B -> A ~~ (A \ {(/)}))
Theorem	limensuci 4506	A limit ordinal is equinumerous to its successor.
|- Lim A   =>   |- (A e. B -> A ~~ suc A)
Theorem	limensuc 4507	A limit ordinal is equinumerous to its successor.
|- ((A e. B /\ Lim A) -> A ~~ suc A)
Pigeonhole Principle
Theorem	phplem1 4508	Lemma for Pigeonhole Principle. If we join a natural number to itself minus an element, we end up with its successor minus the same element.
Theorem	phplem2 4509	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements.
Theorem	phplem3 4510	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor.
Theorem	phplem4 4511	Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers.
Theorem	nneneq 4512	Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136.
|- ((A e. om /\ B e. om) -> (A ~~ B <-> A = B))
Theorem	php 4513	Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of lemmas phplem1 4508 through phplem4 4511, nneneq 4512, and this final piece of the proof.
|- ((A e. om /\ B (. A) -> -. A ~~ B)
Theorem	php2 4514	Corollary of Pigeonhole Principle.
|- ((A e. om /\ B (. A) -> B ~< A)
Theorem	php3 4515	Corollary of Pigeonhole Principle. If A is finite and B is a proper subset of A, the B is strictly less numerous than A. Stronger version of Corollary 6C of [Enderton] p. 135.
|- ((A e. Fin /\ B (. A) -> B ~< A)
Theorem	php3OLD 4516	Corollary of Pigeonhole Principle. If A is finite and B is a proper subset of A, the B is strictly less numerous than A. Stronger version of Corollary 6C of [Enderton] p. 135. (The expression E.x e. omA ~~ x is the definition of "finite," and "infinite" is defined as "not finite.")
|- ((E.x e. om A ~~ x /\ B (. A) -> B ~< A)
Theorem	php4 4517	Corollary of the Pigeonhole Principle php 4513: a natural number is strictly dominated by its successor.
|- (A e. om -> A ~< suc A)
Theorem	php5 4518	Corollary of the Pigeonhole Principle php 4513: a natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90.
|- (A e. om -> -. A ~~ suc A)
Finite sets
Theorem	onomeneq 4519	An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse.
|- ((A e. On /\ B e. om) -> (A ~~ B <-> A = B))
Theorem	onfinOLD 4520	An ordinal number is finite iff it is a natural number. Proposition 10.32 of [TakeutiZaring] p. 92.
|- (A e. On -> (E.x e. om A ~~ x <-> A e. om))
Theorem	nndomo 4521	Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146.
|- ((A e. om /\ B e. om) -> (A ~<_ B <-> A (_ B))
Theorem	nnsdomo 4522	Cardinal ordering agrees with natural number ordering.
|- ((A e. om /\ B e. om) -> (A ~< B <-> A (. B))
Theorem	omsucdom 4523	Strict dominance of natural numbers is the same as dominance over the successor of the smaller.
|- ((A e. om /\ B e. om) -> (A ~< B <-> suc A ~<_ B))
Theorem	sucdomi 4524	Dominance of a set over a successor of a natural number implies strict dominance over the number. For the converse, see sucdom 4842.
|- ((A e. om /\ B e. C) -> (suc A ~<_ B -> A ~< B))
Theorem	0sdom1dom 4525	Strict dominance over zero is the same as dominance over one.
|- A e. V   =>   |- ((/) ~< A <-> 1o ~<_ A)
Theorem	1sdom2 4526	Ordinal 1 is strictly dominated by ordinal 2.
|- 1o ~< 2o
Theorem	finsucdomOLD 4527	Strict dominance of a finite set over a natural number is the same as dominance over its successor.
|- ((A e. om /\ E.x e. om B ~~ x) -> (A ~< B <-> suc A ~<_ B))
Theorem	pssinfOLD 4528	A set equinumerous to a proper subset of itself is infinite. Corollary 6D(a) of [Enderton] p. 136.
|- ((A (. B /\ A ~~ B) -> -. E.x e. om B ~~ x)
Theorem	ominfOLD 4529	The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136.
|- -. E.x e. om om ~~ x
Theorem	omsdomnn 4530	Omega strictly dominates a natural number. Example 3 of [Enderton] p. 146. Here we use A ~<_ om /\ -. om ~~ A instead of A ~< om because, due to a peculiarity ultimately caused our ordered pair definition, we would need the Axiom of infinity (which we have avoided up to now) in order to prove the latter.
|- (A e. om -> (A ~<_ om /\ -. om ~~ A))
Theorem	isfinite1OLD 4531	Omega strictly dominates a finite set. See comment in omsdomnn 4530.
|- (E.x e. om A ~~ x -> (A ~<_ om /\ -. om ~~ A))
Theorem	infsdomnn 4532	An infinite set strictly dominates a natural number.
|- A e. V   =>   |- ((om ~<_ A /\ B e. om) -> B ~< A)
Theorem	infn0 4533	An infinite set is not empty.
|- A e. V   =>   |- (om ~<_ A -> A =/= (/))
Theorem	pssnn 4534	A proper subset of a natural number is equinumerous to some smaller number. Lemma 6F of [Enderton] p. 137.
|- ((A e. om /\ B (. A) -> E.x e. A B ~~ x)
Theorem	ssnnfi 4535	A subset of a natural number is finite.
|- ((A e. om /\ B (_ A) -> B e. Fin)
Theorem	ssnnfiOLD 4536	A subset of a natural number is finite.
|- ((A e. om /\ B (_ A) -> E.x e. om B ~~ x)
Theorem	ssfi 4537	A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138.
|- ((A e. Fin /\ B (_ A) -> B e. Fin)
Theorem	ssfiOLD 4538	A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138.
|- ((E.x e. om A ~~ x /\ B (_ A) -> E.x e. om B ~~ x)
Theorem	domfiOLD 4539	A set dominated by a finite set is finite.
|- ((E.x e. om A ~~ x /\ B ~<_ A) -> E.x e. om B ~~ x)
Theorem	unblem1 4540	Lemma for unbnn 4544. After removing the successor of an element from an unbounded set of natural numbers, the intersection of the result belongs to the original unbounded set.
Theorem	unblem2 4541	Lemma for unbnn 4544. The value of the function F belongs to the unbounded set of natural numbers A.
Theorem	unblem3 4542	Lemma for unbnn 4544. The value of the function F is less than its value at a successor.
Theorem	unblem4 4543	Lemma for unbnn 4544. The function F maps the set of natural numbers one-to-one to the set of unbounded natural numbers A.
Theorem	unbnn 4544	Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. Part of the proof of Theorem 42 of [Suppes] p. 151. See unbnnt 4639 for a stronger version without the hypothesis.
|- A e. V   =>   |- ((A (_ om /\ A.x e. om E.y e. A x e. y) -> A ~~ om)
Theorem	unbnn2 4545	Version of unbnn 4544 that does not require a strict upper bound.
|- A e. V   =>   |- ((A (_ om /\ A.x e. om E.y e. A x (_ y) -> A ~~ om)
Theorem	isfinite2OLD 4546	Any set strictly dominated by the class of natural numbers is finite. Sufficiency part of Theorem 42 of [Suppes] p. 151. This theorem does not require the Axiom of Infinity.
|- (A ~< om -> E.x e. om A ~~ x)
Theorem	fin2inf 4547	This (useless) theorem, which was proved without the Axiom of Infinity, demonstrates an artifact of our definition of strict dominance, which is meaningful only when its arguments exist. In particular, the antecedent cannot be satisfied unless om exists.
|- (A ~< om -> om e. V)
Theorem	unfilem1 4548	Lemma for proving that the union of two finite sets is finite.
Theorem	unfilem2 4549	Lemma for proving that the union of two finite sets is finite.
Theorem	unfilem3 4550	Lemma for proving that the union of two finite sets is finite.
Theorem	unfi 4551	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144.
|- ((A e. Fin /\ B e. Fin) -> (A u. B) e. Fin)
Theorem	unfiOLD 4552	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144.
|- ((E.x e. om A ~~ x /\ E.x e. om B ~~ x) -> E.x e. om (A u. B) ~~ x)
Theorem	unfi2 4553	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 4551 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 4547).
|- ((A ~< om /\ B ~< om) -> (A u. B) ~< om)
Theorem	infcntss 4554	Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.)
|- A e. V   =>   |- (om ~<_ A -> E.x(x (_ A /\ x ~~ om))
Theorem	prfi 4555	An unordered pair is finite.
|- {A, B} e. Fin
Theorem	prfiOLD 4556	An unordered pair is finite.
|- E.x e. om {A, B} ~~ x
Theorem	unifiOLD 4557	The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144.
|- ((E.n e. om A ~~ n /\ A.x e. A E.n e. om x ~~ n) -> E.n e. om U.A ~~ n)
Theorem	unifi2 4558	The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi (future) is useful only if we assume the Axiom of Infinity (see comments in fin2inf 4547).
|- ((A ~< om /\ A.x e. A x ~< om) -> U.A ~< om)
Theorem	fiint 4559	Equivalent ways of stating the finite intersection property. We show two ways of saying, "the intersection of elements in every finite non-empty subcollection of A is in A." This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use the left-hand version of this axiom and others the right-hand version, but as our proof here shows, their "intuitively obvious" equivalence can be non-trivial to establish formally.
|- (A.x e. A A.y e. A (x i^i y) e. A <-> A.x((x (_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A))
Theorem	fiintOLD 4560	Equivalent ways of stating the finite intersection property. We show two ways of saying, "the intersection of elements in every finite non-empty subcollection of A is in A." This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use the left-hand version of this axiom and others the right-hand version, but as our proof here shows, their "intuitively obvious" equivalence can be non-trivial to establish formally.
|- (A.x e. A A.y e. A (x i^i y) e. A <-> A.x((x (_ A /\ x =/= (/) /\ E.y e. om x ~~ y) -> |^|x e. A))
Theorem	abfii1OLD 4561	Two ways to express the collection of finite intersections of a set A.
|- |^|{x | (A (_ x /\ A.y e. x A.z e. x (y i^i z) e. x)} = |^|{x | (A (_ x /\ A.y((y (_ x /\ y =/= (/) /\ E.z e. om y ~~ z) -> |^|y e. x))}
Theorem	abfii2OLD 4562	Two ways to express the collection of finite intersections of a set A.
|- A e. V   =>   |- {x | E.y(y (_ A /\ E.z e. om y ~~ z /\ x = |^|y)} = |^|{x | A.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> |^|y e. x)}
Theorem	abfii3OLD 4563	Two ways to express the collection of finite intersections of a set A.
|- A e. V   =>   |- |^|{x | (A (_ x /\ A.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> |^|y e. x))} = |^|{x | A.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> |^|y e. x)}
Theorem	abfii4OLD 4564	Two ways to express the collection of finite intersections of a set A. Even though the expressions differ by only one symbol, the proof is not simple.
|- A e. V   =>   |- |^|{x | (A (_ x /\ A.y((y (_ x /\ y =/= (/) /\ E.z e. om y ~~ z) -> |^|y e. x))} = |^|{x | (A (_ x /\ A.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> |^|y e. x))}
Theorem	abfii5OLD 4565	Two ways to express the collection of finite intersections of a set A.
|- A e. V   =>   |- |^|{x | (A (_ x /\ A.y e. x A.z e. x (y i^i z) e. x)} =