mmtheorems23 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 2201-2300 - Page 23 of 123

Type	Label	Description
Statement
Theorem	ssnpss 2201	Partial trichotomy law for subclasses.
|- (A (_ B -> -. B (. A)
Theorem	psstr 2202	Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23.
|- ((A (. B /\ B (. C) -> A (. C)
Theorem	sspsstr 2203	Transitive law for subclass and proper subclass.
|- ((A (_ B /\ B (. C) -> A (. C)
Theorem	psssstr 2204	Transitive law for subclass and proper subclass.
|- ((A (. B /\ B (_ C) -> A (. C)
The difference, union, and intersection of two classes
Theorem	difeq1 2205	Equality theorem for class difference.
|- (A = B -> (A \ C) = (B \ C))
Theorem	difeq2 2206	Equality theorem for class difference.
|- (A = B -> (C \ A) = (C \ B))
Theorem	difeq1i 2207	Inference adding difference to the right in a class equality.
|- A = B   =>   |- (A \ C) = (B \ C)
Theorem	difeq2i 2208	Inference adding difference to the left in a class equality.
|- A = B   =>   |- (C \ A) = (C \ B)
Theorem	difeq12i 2209	Equality inference for class difference.
|- A = B   &   |- C = D   =>   |- (A \ C) = (B \ D)
Theorem	difeq1d 2210	Deduction adding difference to the right in a class equality.
|- (ph -> A = B)   =>   |- (ph -> (A \ C) = (B \ C))
Theorem	difeq2d 2211	Deduction adding difference to the left in a class equality.
|- (ph -> A = B)   =>   |- (ph -> (C \ A) = (C \ B))
Theorem	difeqri 2212	Inference from membership to difference.
|- ((x e. A /\ -. x e. B) <-> x e. C)   =>   |- (A \ B) = C
Theorem	hbdif 2213	Bound-variable hypothesis builder for class difference.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. (A \ B) -> A.x y e. (A \ B))
Theorem	eldifi 2214	Implication of membership in a class difference.
|- (A e. (B \ C) -> A e. B)
Theorem	eldifn 2215	Implication of membership in a class difference.
|- (A e. (B \ C) -> -. A e. C)
Theorem	elndif 2216	A set does not belong to a class excluding it.
|- (A e. B -> -. A e. (C \ B))
Theorem	neldif 2217	Implication of membership in a class difference.
|- ((A e. B /\ -. A e. (B \ C)) -> A e. C)
Theorem	difdif 2218	Double class difference. Exercise 11 of [TakeutiZaring] p. 22.
|- (A \ (B \ A)) = A
Theorem	difss 2219	Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22.
|- (A \ B) (_ A
Theorem	ssdifss 2220	Preservation of a subclass relationship by class difference.
|- (A (_ B -> (A \ C) (_ B)
Theorem	ddif 2221	Double complement under universal class. Exercise 4.10(s) of [Mendelson] p. 231.
|- (V \ (V \ A)) = A
Theorem	ssconb 2222	Contraposition law for subsets.
|- ((A (_ C /\ B (_ C) -> (A (_ (C \ B) <-> B (_ (C \ A)))
Theorem	sscon 2223	Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22.
|- (A (_ B -> (C \ B) (_ (C \ A))
Theorem	ssdif 2224	Difference law for subsets.
|- (A (_ B -> (A \ C) (_ (B \ C))
Theorem	elun 2225	Expansion of membership in class union. Theorem 12 of [Suppes] p. 25.
|- (A e. (B u. C) <-> (A e. B \/ A e. C))
Theorem	uneqri 2226	Inference from membership to union.
|- ((x e. A \/ x e. B) <-> x e. C)   =>   |- (A u. B) = C
Theorem	unidm 2227	Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27.
|- (A u. A) = A
Theorem	uncom 2228	Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17.
|- (A u. B) = (B u. A)
Theorem	uneq1 2229	Equality theorem for union of two classes.
|- (A = B -> (A u. C) = (B u. C))
Theorem	uneq2 2230	Equality theorem for the union of two classes.
|- (A = B -> (C u. A) = (C u. B))
Theorem	uneq12 2231	Equality theorem for union of two classes.
|- ((A = B /\ C = D) -> (A u. C) = (B u. D))
Theorem	uneq1i 2232	Inference adding union to the right in a class equality.
|- A = B   =>   |- (A u. C) = (B u. C)
Theorem	uneq2i 2233	Inference adding union to the left in a class equality.
|- A = B   =>   |- (C u. A) = (C u. B)
Theorem	uneq12i 2234	Equality inference for union of two classes. (The proof was shortened by Eric Schmidt, 26-Jan-2007.)
|- A = B   &   |- C = D   =>   |- (A u. C) = (B u. D)
Theorem	uneq1d 2235	Deduction adding union to the right in a class equality.
|- (ph -> A = B)   =>   |- (ph -> (A u. C) = (B u. C))
Theorem	uneq2d 2236	Deduction adding union to the left in a class equality.
|- (ph -> A = B)   =>   |- (ph -> (C u. A) = (C u. B))
Theorem	uneq12d 2237	Equality deduction for union of two classes.
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> (A u. C) = (B u. D))
Theorem	hbun 2238	Bound-variable hypothesis builder for the union of classes.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. (A u. B) -> A.x y e. (A u. B))
Theorem	unass 2239	Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17.
|- ((A u. B) u. C) = (A u. (B u. C))
Theorem	un12 2240	A rearrangement of union.
|- (A u. (B u. C)) = (B u. (A u. C))
Theorem	un23 2241	A rearrangement of union.
|- ((A u. B) u. C) = ((A u. C) u. B)
Theorem	un4 2242	A rearrangement of the union of 4 classes.
|- ((A u. B) u. (C u. D)) = ((A u. C) u. (B u. D))
Theorem	unundi 2243	Union distributes over itself.
|- (A u. (B u. C)) = ((A u. B) u. (A u. C))
Theorem	unundir 2244	Union distributes over itself.
|- ((A u. B) u. C) = ((A u. C) u. (B u. C))
Theorem	ssun1 2245	Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27.
|- A (_ (A u. B)
Theorem	ssun2 2246	Subclass relationship for union of classes.
|- A (_ (B u. A)
Theorem	ssun3 2247	Subclass law for union of classes.
|- (A (_ B -> A (_ (B u. C))
Theorem	ssun4 2248	Subclass law for union of classes.
|- (A (_ B -> A (_ (C u. B))
Theorem	elun1 2249	Membership law for union of classes.
|- (A e. B -> A e. (B u. C))
Theorem	elun2 2250	Membership law for union of classes.
|- (A e. B -> A e. (C u. B))
Theorem	unss1 2251	Subclass law for union of classes.
|- (A (_ B -> (A u. C) (_ (B u. C))
Theorem	ssequn1 2252	A relationship between subclass and union. Theorem 26 of [Suppes] p. 27.
|- (A (_ B <-> (A u. B) = B)
Theorem	unss2 2253	Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18.
|- (A (_ B -> (C u. A) (_ (C u. B))
Theorem	unss12 2254	Subclass law for union of classes.
|- ((A (_ B /\ C (_ D) -> (A u. C) (_ (B u. D))
Theorem	ssequn2 2255	A relationship between subclass and union.
|- (A (_ B <-> (B u. A) = B)
Theorem	unss 2256	The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse.
|- ((A (_ C /\ B (_ C) <-> (A u. B) (_ C)
Theorem	unssi 2257	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 2258	A condition that implies inclusion in the union of two classes.
|- ((A (_ B \/ A (_ C) -> A (_ (B u. C))
Theorem	elin 2259	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 2260	Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17.
|- (A i^i B) = (B i^i A)
Theorem	ineqri 2261	Inference from membership to intersection.
|- ((x e. A /\ x e. B) <-> x e. C)   =>   |- (A i^i B) = C
Theorem	ineq1 2262	Equality theorem for intersection of two classes.
|- (A = B -> (A i^i C) = (B i^i C))
Theorem	ineq2 2263	Equality theorem for intersection of two classes.
|- (A = B -> (C i^i A) = (C i^i B))
Theorem	ineq12 2264	Equality theorem for intersection of two classes.
|- ((A = B /\ C = D) -> (A i^i C) = (B i^i D))
Theorem	ineq1i 2265	Equality inference for intersection of two classes.
|- A = B   =>   |- (A i^i C) = (B i^i C)
Theorem	ineq2i 2266	Equality inference for intersection of two classes.
|- A = B   =>   |- (C i^i A) = (C i^i B)
Theorem	ineq12i 2267	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 2268	Equality deduction for intersection of two classes.
|- (ph -> A = B)   =>   |- (ph -> (A i^i C) = (B i^i C))
Theorem	ineq2d 2269	Equality deduction for intersection of two classes.
|- (ph -> A = B)   =>   |- (ph -> (C i^i A) = (C i^i B))
Theorem	ineq12d 2270	Equality deduction for intersection of two classes.
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> (A i^i C) = (B i^i D))
Theorem	ineqan12d 2271	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 2272	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 2273	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 2274	Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26.
|- (A i^i A) = A
Theorem	inass 2275	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 2276	A rearrangement of intersection.
|- (A i^i (B i^i C)) = (B i^i (A i^i C))
Theorem	in23 2277	A rearrangement of intersection.
|- ((A i^i B) i^i C) = ((A i^i C) i^i B)
Theorem	in4 2278	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 2279	Intersection distributes over itself.
|- (A i^i (B i^i C)) = ((A i^i B) i^i (A i^i C))
Theorem	inindir 2280	Intersection distributes over itself.
|- ((A i^i B) i^i C) = ((A i^i C) i^i (B i^i C))
Theorem	sseqin2 2281	A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18.
|- (A (_ B <-> (B i^i A) = A)
Theorem	inss1 2282	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 2283	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 2284	Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26.
|- ((A (_ B /\ A (_ C) <-> A (_ (B i^i C))
Theorem	ssini 2285	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 2286	Add right intersection to subclass relation.
|- (A (_ B -> (A i^i C) (_ (B i^i C))
Theorem	sslin 2287	Add left intersection to subclass relation.
|- (A (_ B -> (C i^i A) (_ (C i^i B))
Theorem	ss2in 2288	Intersection of subclasses.
|- ((A (_ B /\ C (_ D) -> (A i^i C) (_ (B i^i D))
Theorem	ssinss1 2289	Intersection preserves subclass relationship.
|- (A (_ C -> (A i^i B) (_ C)
Theorem	unabs 2290	Absorption law for union.
|- (A u. (A i^i B)) = A
Theorem	inabs 2291	Absorption law for intersection.
|- (A i^i (A u. B)) = A
Theorem	nssinpss 2292	Negation of subclass expressed in terms of intersection and proper subclass.
|- (-. A (_ B <-> (A i^i B) (. A)
Theorem	nsspssun 2293	Negation of subclass expressed in terms of proper subclass and union.
|- (-. A (_ B <-> B (. (A u. B))
Theorem	dfss4 2294	Subclass defined in terms of class difference. See comments under dfun2 2295.
|- (A (_ B <-> (B \ (B \ A)) = A)
Theorem	dfun2 2295	An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 2296 and dfss4 2294 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 2296	An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 2295. Another version is given by dfin4 2300.
|- (A i^i B) = (A \ (V \ B))
Theorem	difin 2297	Difference with intersection. Theorem 33 of [Suppes] p. 29.
|- (A \ (A i^i B)) = (A \ B)
Theorem	dfun3 2298	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 2299	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 2300	Alternate definition of the intersection of two classes. Exercise 4.10(q) of [Mendelson] p. 231.
|- (A i^i B) = (A \ (A \ B))