mmtheorems23 - Metamath Proof Explorer

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10683

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