mmtheorems23 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 2201-2300 - Page 23 of 107

Type	Label	Description
Statement
Theorem	unssi 2201	An inference that the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
|- A (_ C   &   |- B (_ C   =>   |- (A u. B) (_ C
Theorem	ssun 2202	A condition that implies inclusion in the union of two classes.
|- ((A (_ B \/ A (_ C) -> A (_ (B u. C))
Theorem	elin 2203	Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25.
|- (A e. (B i^i C) <-> (A e. B /\ A e. C))
Theorem	incom 2204	Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17.
|- (A i^i B) = (B i^i A)
Theorem	ineqri 2205	Inference from membership to intersection.
|- ((x e. A /\ x e. B) <-> x e. C)   =>   |- (A i^i B) = C
Theorem	ineq1 2206	Equality theorem for intersection of two classes.
|- (A = B -> (A i^i C) = (B i^i C))
Theorem	ineq2 2207	Equality theorem for intersection of two classes.
|- (A = B -> (C i^i A) = (C i^i B))
Theorem	ineq12 2208	Equality theorem for intersection of two classes.
|- ((A = B /\ C = D) -> (A i^i C) = (B i^i D))
Theorem	ineq1i 2209	Equality inference for intersection of two classes.
|- A = B   =>   |- (A i^i C) = (B i^i C)
Theorem	ineq2i 2210	Equality inference for intersection of two classes.
|- A = B   =>   |- (C i^i A) = (C i^i B)
Theorem	ineq12i 2211	Equality inference for intersection of two classes. (The proof was shortened by Eric Schmidt, 26-Jan-2007.)
|- A = B   &   |- C = D   =>   |- (A i^i C) = (B i^i D)
Theorem	ineq1d 2212	Equality deduction for intersection of two classes.
|- (ph -> A = B)   =>   |- (ph -> (A i^i C) = (B i^i C))
Theorem	ineq2d 2213	Equality deduction for intersection of two classes.
|- (ph -> A = B)   =>   |- (ph -> (C i^i A) = (C i^i B))
Theorem	ineq12d 2214	Equality deduction for intersection of two classes.
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> (A i^i C) = (B i^i D))
Theorem	ineqan12d 2215	Equality deduction for intersection of two classes.
|- (ph -> A = B)   &   |- (ps -> C = D)   =>   |- ((ph /\ ps) -> (A i^i C) = (B i^i D))
Theorem	hbin 2216	Bound-variable hypothesis builder for the intersection of classes.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. (A i^i B) -> A.x y e. (A i^i B))
Theorem	rabbirdv 2217	Deduction from wff to restricted class abstraction.
|- (ph -> (x e. B -> (x e. A <-> ch)))   =>   |- (ph -> (B i^i A) = {x e. B | ch})
Theorem	inidm 2218	Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26.
|- (A i^i A) = A
Theorem	inass 2219	Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17.
|- ((A i^i B) i^i C) = (A i^i (B i^i C))
Theorem	in12 2220	A rearrangement of intersection.
|- (A i^i (B i^i C)) = (B i^i (A i^i C))
Theorem	in23 2221	A rearrangement of intersection.
|- ((A i^i B) i^i C) = ((A i^i C) i^i B)
Theorem	in4 2222	Rearrangement of intersection of 4 classes.
|- ((A i^i B) i^i (C i^i D)) = ((A i^i C) i^i (B i^i D))
Theorem	inindi 2223	Intersection distributes over itself.
|- (A i^i (B i^i C)) = ((A i^i B) i^i (A i^i C))
Theorem	inindir 2224	Intersection distributes over itself.
|- ((A i^i B) i^i C) = ((A i^i C) i^i (B i^i C))
Theorem	sseqin2 2225	A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18.
|- (A (_ B <-> (B i^i A) = A)
Theorem	inss1 2226	The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18.
|- (A i^i B) (_ A
Theorem	inss2 2227	The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18.
|- (A i^i B) (_ B
Theorem	ssin 2228	Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26.
|- ((A (_ B /\ A (_ C) <-> A (_ (B i^i C))
Theorem	ssini 2229	An inference showing that the a subclass of two classes is a subclass of their intersection.
|- A (_ B   &   |- A (_ C   =>   |- A (_ (B i^i C)
Theorem	ssrin 2230	Add right intersection to subclass relation.
|- (A (_ B -> (A i^i C) (_ (B i^i C))
Theorem	sslin 2231	Add left intersection to subclass relation.
|- (A (_ B -> (C i^i A) (_ (C i^i B))
Theorem	ss2in 2232	Intersection of subclasses.
|- ((A (_ B /\ C (_ D) -> (A i^i C) (_ (B i^i D))
Theorem	ssinss1 2233	Intersection preserves subclass relationship.
|- (A (_ C -> (A i^i B) (_ C)
Theorem	unabs 2234	Absorption law for union.
|- (A u. (A i^i B)) = A
Theorem	inabs 2235	Absorption law for intersection.
|- (A i^i (A u. B)) = A
Theorem	nssinpss 2236	Negation of subclass expressed in terms of intersection and proper subclass.
|- (-. A (_ B <-> (A i^i B) (. A)
Theorem	nsspssun 2237	Negation of subclass expressed in terms of proper subclass and union.
|- (-. A (_ B <-> B (. (A u. B))
Theorem	dfss4 2238	Subclass defined in terms of class difference. See comments under dfun2 2239.
|- (A (_ B <-> (B \ (B \ A)) = A)
Theorem	dfun2 2239	An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 2240 and dfss4 2238 it shows we can express union, intersection, and subset directly in terms of the single "primitive" operation \ (class difference).
|- (A u. B) = (V \ ((V \ A) \ B))
Theorem	dfin2 2240	An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 2239. Another version is given by dfin4 2244.
|- (A i^i B) = (A \ (V \ B))
Theorem	difin 2241	Difference with intersection. Theorem 33 of [Suppes] p. 29.
|- (A \ (A i^i B)) = (A \ B)
Theorem	dfun3 2242	Union defined in terms of intersection (DeMorgan's law). Definition of union in [Mendelson] p. 231.
|- (A u. B) = (V \ ((V \ A) i^i (V \ B)))
Theorem	dfin3 2243	Intersection defined in terms of union (DeMorgan's law. Similar to Exercise 4.10(n) of [Mendelson] p. 231.
|- (A i^i B) = (V \ ((V \ A) u. (V \ B)))
Theorem	dfin4 2244	Alternate definition of the intersection of two classes. Exercise 4.10(q) of [Mendelson] p. 231.
|- (A i^i B) = (A \ (A \ B))
Theorem	invdif 2245	Intersection with universal complement. Remark in [Stoll] p. 20.
|- (A i^i (V \ B)) = (A \ B)
Theorem	indif 2246	Intersection with class difference. Theorem 34 of [Suppes] p. 29.
|- (A i^i (A \ B)) = (A \ B)
Theorem	indi 2247	Distributive law for intersection over union. Exercise 10 of [TakeutiZaring] p. 17.
|- (A i^i (B u. C)) = ((A i^i B) u. (A i^i C))
Theorem	undi 2248	Distributive law for union over intersection. Exercise 11 of [TakeutiZaring] p. 17.
|- (A u. (B i^i C)) = ((A u. B) i^i (A u. C))
Theorem	indir 2249	Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27.
|- ((A u. B) i^i C) = ((A i^i C) u. (B i^i C))
Theorem	undir 2250	Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27.
|- ((A i^i B) u. C) = ((A u. C) i^i (B u. C))
Theorem	unineq 2251	Infer equality from equalities of union and intersection. Exercise 20 of [Enderton] p. 32 and its converse.
|- (((A u. C) = (B u. C) /\ (A i^i C) = (B i^i C)) <-> A = B)
Theorem	uneqin 2252	Equality of union and intersection implies equality of their arguments.
|- ((A u. B) = (A i^i B) <-> A = B)
Theorem	difundi 2253	Distributive law for class difference. Theorem 39 of [Suppes] p. 29.
|- (A \ (B u. C)) = ((A \ B) i^i (A \ C))
Theorem	difundir 2254	Distributive law for class difference.
|- ((A u. B) \ C) = ((A \ C) u. (B \ C))
Theorem	difindi 2255	Distributive law for class difference. Theorem 40 of [Suppes] p. 29.
|- (A \ (B i^i C)) = ((A \ B) u. (A \ C))
Theorem	difindir 2256	Distributive law for class difference.
|- ((A i^i B) \ C) = ((A \ C) i^i (B \ C))
Theorem	undm 2257	DeMorgan's law for union. Theorem 5.2(13) of [Stoll] p. 19.
|- (V \ (A u. B)) = ((V \ A) i^i (V \ B))
Theorem	indm 2258	DeMorgan's law for intersection. Theorem 5.2(13') of [Stoll] p. 19.
|- (V \ (A i^i B)) = ((V \ A) u. (V \ B))
Theorem	difun1 2259	A relationship involving double difference and union.
|- (A \ (B u. C)) = ((A \ B) \ C)
Theorem	dif23 2260	Swap second and third argument of double difference.
|- ((A \ B) \ C) = ((A \ C) \ B)
Theorem	symdif1 2261	Two ways to express symmetric difference. This theorem shows the equivalence of the definition of symmetric difference in [Stoll] p. 13 and the restated definition in Example 4.1 of [Stoll] p. 262.
|- ((A \ B) u. (B \ A)) = ((A u. B) \ (A i^i B))
Theorem	symdif2 2262	Two ways to express symmetric difference.
|- ((A \ B) u. (B \ A)) = {x | -. (x e. A <-> x e. B)}
Theorem	unab 2263	Union of two class abstractions.
|- ({x | ph} u. {x | ps}) = {x | (ph \/ ps)}
Theorem	inab 2264	Intersection of two class abstractions.
|- ({x | ph} i^i {x | ps}) = {x | (ph /\ ps)}
Theorem	difab 2265	Difference of two class abstractions.
|- ({x | ph} \ {x | ps}) = {x | (ph /\ -. ps)}
Theorem	unrab 2266	Union of two restricted class abstractions.
|- ({x e. A | ph} u. {x e. A | ps}) = {x e. A | (ph \/ ps)}
Theorem	inrab 2267	Intersection of two restricted class abstractions.
|- ({x e. A | ph} i^i {x e. A | ps}) = {x e. A | (ph /\ ps)}
Theorem	inrab2 2268	Intersection with a restricted class abstraction.
|- ({x e. A | ph} i^i B) = {x e. (A i^i B) | ph}
Theorem	difrab 2269	Difference of two restricted class abstractions.
|- ({x e. A | ph} \ {x e. A | ps}) = {x e. A | (ph /\ -. ps)}
Theorem