mmtheorems22 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 2101-2200 - Page 22 of 107

Type	Label	Description
Statement
Theorem	syl5ss 2101	A chained subclass and equality deduction.
|- (ph -> A (_ B)   &   |- C = A   =>   |- (ph -> C (_ B)
Theorem	syl5ssr 2102	A chained subclass and equality deduction.
|- (ph -> A (_ B)   &   |- A = C   =>   |- (ph -> C (_ B)
Theorem	syl6ss 2103	A chained subclass and equality deduction.
|- (ph -> A (_ B)   &   |- B = C   =>   |- (ph -> A (_ C)
Theorem	syl6ssr 2104	A chained subclass and equality deduction.
|- (ph -> A (_ B)   &   |- C = B   =>   |- (ph -> A (_ C)
Theorem	eqimss 2105	Equality implies the subclass relation.
|- (A = B -> A (_ B)
Theorem	eqimss2 2106	Equality implies the subclass relation.
|- (B = A -> A (_ B)
Theorem	eqimssi 2107	Infer subclass relationship from equality.
|- A = B   =>   |- A (_ B
Theorem	eqimss2i 2108	Infer subclass relationship from equality.
|- A = B   =>   |- B (_ A
Theorem	nss 2109	Negation of subclass relationship. Exercise 13 of [TakeutiZaring] p. 18.
|- (-. A (_ B <-> E.x(x e. A /\ -. x e. B))
Theorem	ssralv 2110	Quantification restricted to a subclass.
|- (A (_ B -> (A.x e. B ph -> A.x e. A ph))
Theorem	ssrexv 2111	Existential quantification restricted to a subclass.
|- (A (_ B -> (E.x e. A ph -> E.x e. B ph))
Theorem	ss2ab 2112	Class abstractions in a subclass relationship.
|- ({x | ph} (_ {x | ps} <-> A.x(ph -> ps))
Theorem	abss 2113	Class abstraction in a subclass relationship.
|- ({x | ph} (_ A <-> A.x(ph -> x e. A))
Theorem	ssab 2114	Subclass of a class abstraction.
|- (A (_ {x | ph} <-> A.x(x e. A -> ph))
Theorem	ssabral 2115	The relation for a subclass of a class abstraction is equivalent to restricted quantification.
|- (A (_ {x | ph} <-> A.x e. A ph)
Theorem	ss2abi 2116	Inference of abstraction subclass from implication.
|- (ph -> ps)   =>   |- {x | ph} (_ {x | ps}
Theorem	abssdv 2117	Deduction of abstraction subclass from implication.
|- (ph -> (ps -> x e. A))   =>   |- (ph -> {x | ps} (_ A)
Theorem	abssi 2118	Inference of abstraction subclass from implication.
|- (ph -> x e. A)   =>   |- {x | ph} (_ A
Theorem	ss2rab 2119	Restricted abstraction classes in a subclass relationship.
|- ({x e. A | ph} (_ {x e. A | ps} <-> A.x e. A (ph -> ps))
Theorem	rabss 2120	Restricted class abstraction in a subclass relationship.
|- ({x e. A | ph} (_ B <-> A.x e. A (ph -> x e. B))
Theorem	ssrab 2121	Subclass of a restricted class abstraction.
|- (B (_ {x e. A | ph} <-> (B (_ A /\ A.x e. B ph))
Theorem	ssrabdv 2122	Subclass of a restricted class abstraction (deduction rule).
|- (ph -> B (_ A)   &   |- ((ph /\ x e. B) -> ps)   =>   |- (ph -> B (_ {x e. A | ps})
Theorem	ss2rabdv 2123	Deduction of restricted abstraction subclass from implication.
|- ((ph /\ x e. A) -> (ps -> ch))   =>   |- (ph -> {x e. A | ps} (_ {x e. A | ch})
Theorem	ss2rabi 2124	Inference of restricted abstraction subclass from implication.
|- (x e. A -> (ph -> ps))   =>   |- {x e. A | ph} (_ {x e. A | ps}
Theorem	rabss2 2125	Subclass law for restricted abstraction.
|- (A (_ B -> {x e. A | ph} (_ {x e. B | ph})
Theorem	ssab2 2126	Subclass relation for the restriction of a class abstraction.
|- {x | (x e. A /\ ph)} (_ A
Theorem	ssrab2 2127	Subclass relation for a restricted class.
|- {x e. A | ph} (_ A
Theorem	uniiunlem 2128	A subset relationship useful for converting union to indexed union using dfiun2 2583 or dfiun2g 2582 and intersection to indexed intersection using dfiin2 2584.
|- (A.x e. A B e. D -> (A.x e. A B e. C <-> {y | E.x e. A y = B} (_ C))
Theorem	dfpss2 2129	Alternate definition of proper subclass.
|- (A (. B <-> (A (_ B /\ -. A = B))
Theorem	dfpss3 2130	Alternate definition of proper subclass.
|- (A (. B <-> (A (_ B /\ -. B (_ A))
Theorem	psseq1 2131	Equality theorem for proper subclass.
|- (A = B -> (A (. C <-> B (. C))
Theorem	psseq2 2132	Equality theorem for proper subclass.
|- (A = B -> (C (. A <-> C (. B))
Theorem	psseq1i 2133	An equality inference for the proper subclass relationship.
|- A = B   =>   |- (A (. C <-> B (. C)
Theorem	psseq2i 2134	An equality inference for the proper subclass relationship.
|- A = B   =>   |- (C (. A <-> C (. B)
Theorem	psseq12i 2135	An equality inference for the proper subclass relationship.
|- A = B   &   |- C = D   =>   |- (A (. C <-> B (. D)
Theorem	psseq1d 2136	An equality deduction for the proper subclass relationship.
|- (ph -> A = B)   =>   |- (ph -> (A (. C <-> B (. C))
Theorem	psseq2d 2137	An equality deduction for the proper subclass relationship.
|- (ph -> A = B)   =>   |- (ph -> (C (. A <-> C (. B))
Theorem	psseq12d 2138	An equality deduction for the proper subclass relationship.
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> (A (. C <-> B (. D))
Theorem	pssss 2139	A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23.
|- (A (. B -> A (_ B)
Theorem	pssssd 2140	Deduce subclass from proper subclass.
|- (ph -> A (. B)   =>   |- (ph -> A (_ B)
Theorem	sspss 2141	Subclass in terms of proper subclass.
|- (A (_ B <-> (A (. B \/ A = B))
Theorem	pssirr 2142	Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23.
|- -. A (. A
Theorem	pssn2lp 2143	Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23.
|- -. (A (. B /\ B (. A)
Theorem	sspsstri 2144	Two ways of stating trichotomy with respect to inclusion.
|- ((A (_ B \/ B (_ A) <-> (A (. B \/ A = B \/ B (. A))
Theorem	ssnpss 2145	Partial trichotomy law for subclasses.
|- (A (_ B -> -. B (. A)
Theorem	psstr 2146	Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23.
|- ((A (. B /\ B (. C) -> A (. C)
Theorem	sspsstr 2147	Transitive law for subclass and proper subclass.
|- ((A (_ B /\ B (. C) -> A (. C)
Theorem	psssstr 2148	Transitive law for subclass and proper subclass.
|- ((A (. B /\ B (_ C) -> A (. C)
The difference, union, and intersection of two classes
Theorem	difeq1 2149	Equality theorem for class difference.
|- (A = B -> (A \ C) = (B \ C))
Theorem	difeq2 2150	Equality theorem for class difference.
|- (A = B -> (C \ A) = (C \ B))
Theorem	difeq1i 2151	Inference adding difference to the right in a class equality.
|- A = B   =>   |- (A \ C) = (B \ C)
Theorem	difeq2i 2152	Inference adding difference to the left in a class equality.
|- A = B   =>   |- (C \ A) = (C \ B)
Theorem	difeq12i 2153	Equality inference for class difference.
|- A = B   &   |- C = D   =>   |- (A \ C) = (B \ D)
Theorem	difeq1d 2154	Deduction adding difference to the right in a class equality.
|- (ph -> A = B)   =>   |- (ph -> (A \ C) = (B \ C))
Theorem	difeq2d 2155	Deduction adding difference to the left in a class equality.
|- (ph -> A = B)   =>   |- (ph -> (C \ A) = (C \ B))
Theorem	difeqri 2156	Inference from membership to difference.
|- ((x e. A /\ -. x e. B) <-> x e. C)   =>   |- (A \ B) = C
Theorem	hbdif 2157	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 2158	Implication of membership in a class difference.
|- (A e. (B \ C) -> A e. B)
Theorem	eldifn 2159	Implication of membership in a class difference.
|- (A e. (B \ C) -> -. A e. C)
Theorem	elndif 2160	A set does not belong to a class excluding it.
|- (A e. B -> -. A e. (C \ B))
Theorem	neldif 2161	Implication of membership in a class difference.
|- ((A e. B /\ -. A e. (B \ C)) -> A e. C)
Theorem	difdif 2162	Double class difference. Exercise 11 of [TakeutiZaring] p. 22.
|- (A \ (B \ A)) = A
Theorem	difss 2163	Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22.
|- (A \ B) (_ A
Theorem	ssdifss 2164	Preservation of a subclass relationship by class difference.
|- (A (_ B -> (A \ C) (_ B)
Theorem	ddif 2165	Double complement under universal class. Exercise 4.10(s) of [Mendelson] p. 231.
|- (V \ (V \ A)) = A
Theorem	ssconb 2166	Contraposition law for subsets.
|- ((A (_ C /\ B (_ C) -> (A (_ (C \ B) <-> B (_ (C \ A)))
Theorem	sscon 2167	Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22.
|- (A (_ B -> (C \ B) (_ (C \ A))
Theorem	ssdif 2168	Difference law for subsets.
|- (A (_ B -> (A \ C) (_ (B \ C))
Theorem	elun 2169	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 2170	Inference from membership to union.
|- ((x e. A \/ x e. B) <-> x e. C)   =>   |- (A u. B) = C
Theorem	unidm 2171	Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27.
|- (A u. A) = A
Theorem	uncom 2172	Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17.
|- (A u. B) = (B u. A)
Theorem	uneq1 2173	Equality theorem for union of two classes.
|- (A = B -> (A u. C) = (B u. C))
Theorem	uneq2 2174	Equality theorem for the union of two classes.
|- (A = B -> (C u. A) = (C u. B))
Theorem	uneq12 2175	Equality theorem for union of two classes.
|- ((A = B /\ C = D) -> (A u. C) = (B u. D))
Theorem	uneq1i 2176	Inference adding union to the right in a class equality.
|- A = B   =>   |- (A u. C)