mmtheorems15 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 1401-1500 - Page 15 of 107

Type	Label	Description
Statement
Theorem	mo4f 1401	"At most one" expressed using implicit substitution.
|- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- (E*xph <-> A.xA.y((ph /\ ps) -> x = y))
Theorem	mo4 1402	"At most one" expressed using implicit substitution.
|- (x = y -> (ph <-> ps))   =>   |- (E*xph <-> A.xA.y((ph /\ ps) -> x = y))
Theorem	mobid 1403	Formula-building rule for "at most one" quantifier (deduction rule).
|- (ph -> A.xph)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> (E*xps <-> E*xch))
Theorem	mobii 1404	Formula-building rule for "at most one" quantifier (inference rule).
|- (ps <-> ch)   =>   |- (E*xps <-> E*xch)
Theorem	hbmo1 1405	Bound-variable hypothesis builder for "at most one."
|- (E*xph -> A.xE*xph)
Theorem	hbmo 1406	Bound-variable hypothesis builder for "at most one."
|- (ph -> A.xph)   =>   |- (E*yph -> A.xE*yph)
Theorem	cbvmo 1407	Rule used to change bound variables with implicit substitution.
|- (ph -> A.yph)   &   |- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- (E*xph <-> E*yps)
Theorem	eu5 1408	Uniqueness in terms of "at most one."
|- (E!xph <-> (E.xph /\ E*xph))
Theorem	eu4 1409	Uniqueness using implicit substitution.
|- (x = y -> (ph <-> ps))   =>   |- (E!xph <-> (E.xph /\ A.xA.y((ph /\ ps) -> x = y)))
Theorem	eumo 1410	Existential uniqueness implies "at most one."
|- (E!xph -> E*xph)
Theorem	eumoi 1411	"At most one" inferred from existential uniqueness.
|- E!xph   =>   |- E*xph
Theorem	exmoeu 1412	Existence in terms of "at most one" and uniqueness.
|- (E.xph <-> (E*xph -> E!xph))
Theorem	exmoeu2 1413	Existence implies "at most one" is equivalent to uniqueness.
|- (E.xph -> (E*xph <-> E!xph))
Theorem	moabs 1414	Absorption of existence condition by "at most one."
|- (E*xph <-> (E.xph -> E*xph))
Theorem	exmo 1415	Something exists or at most one exists.
|- (E.xph \/ E*xph)
Theorem	immo 1416	"At most one" is preserved through implication (notice wff reversal).
|- (A.x(ph -> ps) -> (E*xps -> E*xph))
Theorem	immoi 1417	"At most one" is preserved through implication (notice wff reversal).
|- (ph -> ps)   =>   |- (E*xps -> E*xph)
Theorem	moimv 1418	Move antecedent outside of "at most one."
|- (E*x(ph -> ps) -> (ph -> E*xps))
Theorem	euimmo 1419	Uniqueness implies "at most one" through implication.
|- (A.x(ph -> ps) -> (E!xps -> E*xph))
Theorem	euim 1420	Add existential uniqueness quantifiers to an implication. Note the reversed implication in the antecedent.
|- ((E.xph /\ A.x(ph -> ps)) -> (E!xps -> E!xph))
Theorem	moan 1421	"At most one" is still the case when a conjunct is added.
|- (E*xph -> E*x(ps /\ ph))
Theorem	moani 1422	"At most one" is still true when a conjunct is added.
|- E*xph   =>   |- E*x(ps /\ ph)
Theorem	moor 1423	"At most one" is still the case when a disjunct is removed.
|- (E*x(ph \/ ps) -> E*xph)
Theorem	mooran1 1424	"At most one" imports disjunction to conjunction.
|- ((E*xph \/ E*xps) -> E*x(ph /\ ps))
Theorem	mooran2 1425	"At most one" exports disjunction to conjunction.
|- (E*x(ph \/ ps) -> (E*xph /\ E*xps))
Theorem	moanim 1426	Introduction of a conjunct into "at most one" quantifier.
|- (ph -> A.xph)   =>   |- (E*x(ph /\ ps) <-> (ph -> E*xps))
Theorem	euan 1427	Introduction of a conjunct into uniqueness quantifier.
|- (ph -> A.xph)   =>   |- (E!x(ph /\ ps) <-> (ph /\ E!xps))
Theorem	moanimv 1428	Introduction of a conjunct into "at most one" quantifier.
|- (E*x(ph /\ ps) <-> (ph -> E*xps))
Theorem	moaneu 1429	Nested "at most one" and uniqueness quantifiers.
|- E*x(ph /\ E!xph)
Theorem	moanmo 1430	Nested "at most one" quantifiers.
|- E*x(ph /\ E*xph)
Theorem	euanv 1431	Introduction of a conjunct into uniqueness quantifier.
|- (E!x(ph /\ ps) <-> (ph /\ E!xps))
Theorem	mopick 1432	"At most one" picks a variable value, eliminating an existential quantifier.
|- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))
Theorem	eupick 1433	Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing x such that ph is true, and there is also an x (actually the same one) such that ph and ps are both true, then ph implies ps regardless of x. This theorem, which apparently does not appear explicitly in the literature, can be quite useful because it lets us eliminate existential quantifiers in a hypothesis.
|- ((E!xph /\ E.x(ph /\ ps)) -> (ph -> ps))
Theorem	eupickb 1434	Existential uniqueness "pick" showing wff equivalence.
|- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (ph <-> ps))
Theorem	mopick2 1435	"At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1093.
|- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> E.x(ph /\ ps /\ ch))
Theorem	euor2 1436	Introduce or eliminate a disjunct in a uniqueness quantifier.
|- (-. E.xph -> (E!x(ph \/ ps) <-> E!xps))
Theorem	moexex 1437	"At most one" double quantification.
|- (ph -> A.yph)   =>   |- ((E*xph /\ A.xE*yps) -> E*yE.x(ph /\ ps))
Theorem	moexexv 1438	"At most one" double quantification.
|- ((E*xph /\ A.xE*yps) -> E*yE.x(ph /\ ps))
Theorem	2moex 1439	Double quantification with "at most one."
|- (E*xE.yph -> A.yE*xph)
Theorem	2euex 1440	Double quantification with existential uniqueness.
|- (E!xE.yph -> E.yE!xph)
Theorem	2eumo 1441	Double quantification with existential uniqueness and "at most one."
|- (E!xE*yph -> E*xE!yph)
Theorem	2eu2ex 1442	Double existential uniqueness.
|- (E!xE!yph -> E.xE.yph)
Theorem	2moswap 1443	A condition allowing swap of "at most one" and existential quantifiers.
|- (A.xE*yph -> (E*xE.yph -> E*yE.xph))
Theorem	2euswap 1444	A condition allowing swap of uniqueness and existential quantifiers.
|- (A.xE*yph -> (E!xE.yph -> E!yE.xph))
Theorem	2exeu 1445	Double existential uniqueness implies double uniqueness quantification.
|- ((E!xE.yph /\ E!yE.xph) -> E!xE!yph)
Theorem	2mo 1446	Two equivalent expressions for double "at most one."
|- (E.zE.wA.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)))
Theorem	2mos 1447	Double "exists at most one" with implicit substitution.
|- ((x = z /\ y = w) -> (ph <-> ps))   =>   |- (E.zE.wA.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.yA.zA.w((ph /\ ps) -> (x = z /\ y = w)))
Theorem	2eu1 1448	Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one.
|- (A.xE*yph -> (E!xE!yph <-> (E!xE.yph /\ E!yE.xph)))
Theorem	2eu2 1449	Double existential uniqueness.
|- (E!yE.xph -> (E!xE!yph <-> E!xE.yph))
Theorem	2eu3 1450	Double existential uniqueness.
|- (A.xA.y(E*xph \/ E*yph) -> ((E!xE!yph /\ E!yE!xph) <-> (E!xE.yph /\ E!yE.xph)))
Theorem	2eu4 1451	This theorem provides us with a definition of double existential uniqueness ("exactly one x and exactly one y"). Naively one might think (incorrectly) that it could be defined by E!xE!yph. See 2eu1 1448 for a condition under which the naive definition holds and 2exeu 1445 for a one-way implication. See 2eu5 1452 and 2eu8 1455 for alternate definitions.
|- ((E!xE.yph /\ E!yE.xph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))
Theorem	2eu5 1452	An alternate definition of double existential uniqueness (see 2eu4 1451). A mistake sometimes made in the literature is to use E!xE!y to mean "exactly one x and exactly one y." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining A.xE*yph as an additional condition. The correct definition apparently has never been published. (E* means "exists at most one.")
|- ((E!xE!yph /\ A.xE*yph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))
Theorem	2eu6 1453	Two equivalent expressions for double existential uniqueness.
|- ((E!xE.yph /\ E!yE.xph) <-> E.zE.wA.xA.y(ph <-> (x = z /\ y = w)))
Theorem	2eu7 1454	Two equivalent expressions for double existential uniqueness.
|- ((E!xE.yph /\ E!yE.xph) <-> E!xE!y(E.xph /\ E.yph))
Theorem	2eu8 1455	Two equivalent expressions for double existential uniqueness. Curiously, we can put E! on either of the internal conjuncts but not both. We can also commute E!xE!y using 2eu7 1454.
|- (E!xE!y(E.xph /\ E.yph) <-> E!xE!y(E!xph /\ E.yph))
Theorem	exists1 1456	Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru 2768.
|- (E!x x = x <-> A.x x = y)
Theorem	exists2 1457	A condition implying that at least two things exist.
|- ((E.xph /\ E.x -. ph) -> -. E!x x = x)
ZF Set Theory - start with the Axiom of Extensionality
Introduce the Axiom of Extensionality
Axiom	ax-ext 1458	Axiom of Extensionality. An axiom of Zermelo-Fraenkel set theory. It states that two sets are identical if they contain the same elements. Axiom Ext of [BellMachover] p. 461. Set theory can also be formulated with a single primitive predicate e. on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes (A.w(w e. x <-> w e. y) -> (x e. z -> y e. z)), and equality x = y is defined as A.w(w e. x <-> w e. y). All of the usual axioms of equality then become theorems of set theory. See, for example, Axiom 1 of [TakeutiZaring] p. 8. To use the above "equality-free" version of Extensionality with Metamath's logical axioms, we would rewrite ax-8 963 through ax-16 1209 with equality expanded according to the above definition. Some of those axioms could be proved from set theory and would be redundant. Not all of them are redundant, since our axioms of predicate calculus make essential use of equality for the proper substitution that is a primitive notion in traditional predicate calculus. A study of such an axiomatization would be an interesting project for someone exploring the foundations of logic. General remarks: Our set theory axioms are presented using defined connectives (<->, E., etc.) for convenience. However, it is implicitly understood that the actual axioms use only the primitive connectives ->, -., A., =, and e.. It is straightforward to establish the equivalence between the actual axioms and the ones we display, and we will not do so. It is important to understand that strictly speaking, all of our set theory axioms are really schemes that represent an infinite number of actual axioms. This is inherent in the design of Metamath ("metavariable math"), which manipulates only metavariables. For example, the metavariable x in ax-ext 1458 can represent any actual variable v1, v2, v3,... . Distinct variable restrictions ($d) prevent us from substituting say v1 for both x and z. This is in contrast to typical textbook presentations that present actual axioms (except for Replacement ax-rep 2689, which involves a wff metavariable). In practice, though, the theorems and proofs are essentially the same. The $d restrictions make each of the the infinite axioms generated by the ax-ext 1458 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version.
|- (A.z(z e. x <-> z e. y) -> x = y)
Theorem