mmtheorems22 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 2101-2200 - Page 22 of 123

Type	Label	Description
Statement
Definition	df-dif 2101	Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Several notations are used in the literature; we chose the \ convention used in Definition 5.3 of [Eisenberg] p. 67 instead of the more common minus sign to reserve the latter for later use in, e.g., arithmetic. We will use the terminology "A excludes B" to mean A \ B. We will use "B is removed from A" to mean A \ {B} i.e. the removal of an element or equivalently the exclusion of a singleton.
|- (A \ B) = {x | (x e. A /\ -. x e. B)}
Definition	df-un 2102	Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For an alternate definition in terms of class difference, requiring no dummy variables, see dfun2 2295. For union defined in terms of intersection, see dfun3 2298.
|- (A u. B) = {x | (x e. A \/ x e. B)}
Definition	df-in 2103	Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 2296 and dfin4 2300. For intersection defined in terms of union, see dfin3 2299.
|- (A i^i B) = {x | (x e. A /\ x e. B)}
Theorem	dfin5 2104	Alternate definition for the intersection of two classes.
|- (A i^i B) = {x e. A | x e. B}
Definition	df-ss 2105	Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. For a more traditional definition, but requiring a dummy variable, see dfss2 2110. Other possible definitions are given by dfss3 2111, dfss4 2294, sspss 2197, ssequn1 2252, ssequn2 2255, sseqin2 2281, and ssdif0 2380.
|- (A (_ B <-> (A i^i B) = A)
Theorem	dfss 2106	A frequently-used variant of subclass definition df-ss 2105.
|- (A (_ B <-> A = (A i^i B))
Definition	df-pss 2107	Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. Other possible definitions are given by dfpss2 2185 and dfpss3 2186.
|- (A (. B <-> (A (_ B /\ A =/= B))
Theorem	dfdif2 2108	Alternate definition of class difference.
|- (A \ B) = {x e. A | -. x e. B}
Theorem	eldif 2109	Expansion of membership in a class difference.
|- (A e. (B \ C) <-> (A e. B /\ -. A e. C))
Subclasses and subsets
Theorem	dfss2 2110	Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17.
|- (A (_ B <-> A.x(x e. A -> x e. B))
Theorem	dfss3 2111	Alternate definition of subclass relationship.
|- (A (_ B <-> A.x e. A x e. B)
Theorem	dfss2f 2112	Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (A (_ B <-> A.x(x e. A -> x e. B))
Theorem	dfss3f 2113	Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (A (_ B <-> A.x e. A x e. B)
Theorem	hbss 2114	If x is not free in A and B, it is not free in A (_ B.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (A (_ B -> A.x A (_ B)
Theorem	ssel 2115	Membership relationships follow from a subclass relationship.
|- (A (_ B -> (C e. A -> C e. B))
Theorem	ssel2 2116	Membership relationships follow from a subclass relationship.
|- ((A (_ B /\ C e. A) -> C e. B)
Theorem	sseli 2117	Membership inference from subclass relationship.
|- A (_ B   =>   |- (C e. A -> C e. B)
Theorem	sselii 2118	Membership inference from subclass relationship.
|- A (_ B   &   |- C e. A   =>   |- C e. B
Theorem	sseld 2119	Membership deduction from subclass relationship.
|- (ph -> A (_ B)   =>   |- (ph -> (C e. A -> C e. B))
Theorem	sseldd 2120	Membership inference from subclass relationship.
|- (ph -> A (_ B)   &   |- (ph -> C e. A)   =>   |- (ph -> C e. B)
Theorem	ssriv 2121	Inference rule based on subclass definition.
|- (x e. A -> x e. B)   =>   |- A (_ B
Theorem	ssrdv 2122	Deduction rule based on subclass definition.
|- (ph -> (x e. A -> x e. B))   =>   |- (ph -> A (_ B)
Theorem	sstr2 2123	Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17.
|- (A (_ B -> (B (_ C -> A (_ C))
Theorem	sstr 2124	Transitivity of subclasses. Theorem 6 of [Suppes] p. 23.
|- ((A (_ B /\ B (_ C) -> A (_ C)
Theorem	sstri 2125	Subclass transitivity inference.
|- A (_ B   &   |- B (_ C   =>   |- A (_ C
Theorem	sstrd 2126	Subclass transitivity deduction.
|- (ph -> A (_ B)   &   |- (ph -> B (_ C)   =>   |- (ph -> A (_ C)
Theorem	sylan9ss 2127	A subclass transitivity deduction.
|- (ph -> A (_ B)   &   |- (ps -> B (_ C)   =>   |- ((ph /\ ps) -> A (_ C)
Theorem	sylan9ssr 2128	A subclass transitivity deduction.
|- (ph -> A (_ B)   &   |- (ps -> B (_ C)   =>   |- ((ps /\ ph) -> A (_ C)
Theorem	eqss 2129	The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22.
|- (A = B <-> (A (_ B /\ B (_ A))
Theorem	eqssi 2130	Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22.
|- A (_ B   &   |- B (_ A   =>   |- A = B
Theorem	eqssd 2131	Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22.
|- (ph -> A (_ B)   &   |- (ph -> B (_ A)   =>   |- (ph -> A = B)
Theorem	ssid 2132	Any class is a subclass of itself. Exercise 10 of [TakeutiZaring] p. 18.
|- A (_ A
Theorem	ssv 2133	Any class is a subclass of the universal class.
|- A (_ V
Theorem	sseq1 2134	Equality theorem for subclasses.
|- (A = B -> (A (_ C <-> B (_ C))
Theorem	sseq2 2135	Equality theorem for the subclass relationship.
|- (A = B -> (C (_ A <-> C (_ B))
Theorem	sseq12 2136	Equality theorem for the subclass relationship.
|- ((A = B /\ C = D) -> (A (_ C <-> B (_ D))
Theorem	sseq1i 2137	An equality inference for the subclass relationship.
|- A = B   =>   |- (A (_ C <-> B (_ C)
Theorem	sseq2i 2138	An equality inference for the subclass relationship.
|- A = B   =>   |- (C (_ A <-> C (_ B)
Theorem	sseq12i 2139	An equality inference for the subclass relationship. (The proof was shortened by Eric Schmidt, 26-Jan-2007.)
|- A = B   &   |- C = D   =>   |- (A (_ C <-> B (_ D)
Theorem	sseq1d 2140	An equality deduction for the subclass relationship.
|- (ph -> A = B)   =>   |- (ph -> (A (_ C <-> B (_ C))
Theorem	sseq2d 2141	An equality deduction for the subclass relationship.
|- (ph -> A = B)   =>   |- (ph -> (C (_ A <-> C (_ B))
Theorem	sseq12d 2142	An equality deduction for the subclass relationship.
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> (A (_ C <-> B (_ D))
Theorem	eqsstri 2143	Substitution of equality into a subclass relationship.
|- A = B   &   |- B (_ C   =>   |- A (_ C
Theorem	eqsstr3i 2144	Substitution of equality into a subclass relationship.
|- B = A   &   |- B (_ C   =>   |- A (_ C
Theorem	sseqtri 2145	Substitution of equality into a subclass relationship.
|- A (_ B   &   |- B = C   =>   |- A (_ C
Theorem	sseqtr4i 2146	Substitution of equality into a subclass relationship.
|- A (_ B   &   |- C = B   =>   |- A (_ C
Theorem	eqsstrd 2147	Substitution of equality into a subclass relationship.
|- (ph -> A = B)   &   |- (ph -> B (_ C)   =>   |- (ph -> A (_ C)
Theorem	eqsstr3d 2148	Substitution of equality into a subclass relationship.
|- (ph -> B = A)   &   |- (ph -> B (_ C)   =>   |- (ph -> A (_ C)
Theorem	sseqtrd 2149	Substitution of equality into a subclass relationship.
|- (ph -> A (_ B)   &   |- (ph -> B = C)   =>   |- (ph -> A (_ C)
Theorem	sseqtr4d 2150	Substitution of equality into a subclass relationship.
|- (ph -> A (_ B)   &   |- (ph -> C = B)   =>   |- (ph -> A (_ C)
Theorem	3sstr3i 2151	Substitution of equality in both sides of a subclass relationship. (The proof was shortened by Eric Schmidt, 26-Jan-2007.)
|- A (_ B   &   |- A = C   &   |- B = D   =>   |- C (_ D
Theorem	3sstr4i 2152	Substitution of equality in both sides of a subclass relationship. (The proof was shortened by Eric Schmidt, 26-Jan-2007.)
|- A (_ B   &   |- C = A   &   |- D = B   =>   |- C (_ D
Theorem	3sstr3g 2153	Substitution of equality into both sides of a subclass relationship.
|- (ph -> A (_ B)   &   |- A = C   &   |- B = D   =>   |- (ph -> C (_ D)
Theorem	3sstr4g 2154	Substitution of equality into both sides of a subclass relationship. (The proof was shortened by Eric Schmidt, 26-Jan-2007.)
|- (ph -> A (_ B)   &   |- C = A   &   |- D = B   =>   |- (ph -> C (_ D)
Theorem	3sstr3d 2155	Substitution of equality into both sides of a subclass relationship.
|- (ph -> A (_ B)   &   |- (ph -> A = C)   &   |- (ph -> B = D)   =>   |- (ph -> C (_ D)
Theorem	3sstr4d 2156	Substitution of equality into both sides of a subclass relationship. (The proof was shortened by Eric Schmidt, 26-Jan-2007.)
|- (ph -> A (_ B)   &   |- (ph -> C = A)   &   |- (ph -> D = B)   =>   |- (ph -> C (_ D)
Theorem	syl5ss 2157	A chained subclass and equality deduction.
|- (ph -> A (_ B)   &   |- C = A   =>   |- (ph -> C (_ B)
Theorem	syl5ssr 2158	A chained subclass and equality deduction.
|- (ph -> A (_ B)   &   |- A = C   =>   |- (ph -> C (_ B)
Theorem	syl6ss 2159	A chained subclass and equality deduction.
|- (ph -> A (_ B)   &   |- B = C   =>   |- (ph -> A (_ C)
Theorem	syl6ssr 2160	A chained subclass and equality deduction.
|- (ph -> A (_ B)   &   |- C = B   =>   |- (ph -> A (_ C)
Theorem	eqimss 2161	Equality implies the subclass relation.
|- (A = B -> A (_ B)
Theorem	eqimss2 2162	Equality implies the subclass relation.
|- (B = A -> A (_ B)
Theorem	eqimssi 2163	Infer subclass relationship from equality.
|- A = B   =>   |- A (_ B
Theorem	eqimss2i 2164	Infer subclass relationship from equality.
|- A = B   =>   |- B (_ A
Theorem	nss 2165	Negation of subclass relationship. Exercise 13 of [TakeutiZaring] p. 18.
|- (-. A (_ B <-> E.x(x e. A /\ -. x e. B))
Theorem	ssralv 2166	Quantification restricted to a subclass.
|- (A (_ B -> (A.x e. B ph -> A.x e. A ph))
Theorem	ssrexv 2167	Existential quantification restricted to a subclass.
|- (A (_ B -> (E.x e. A ph -> E.x e. B ph))
Theorem	ss2ab 2168	Class abstractions in a subclass relationship.
|- ({x | ph} (_ {x | ps} <-> A.x(ph -> ps))
Theorem	abss 2169	Class abstraction in a subclass relationship.
|- ({x | ph} (_ A <-> A.x(ph -> x e. A))
Theorem	ssab 2170	Subclass of a class abstraction.
|- (A (_ {x | ph} <-> A.x(x e. A -> ph))
Theorem	ssabral 2171	The relation for a subclass of a class abstraction is equivalent to restricted quantification.
|- (A (_ {x | ph} <-> A.x e. A ph)
Theorem	ss2abi 2172	Inference of abstraction subclass from implication.
|- (ph -> ps)   =>   |- {x | ph} (_ {x | ps}
Theorem	abssdv 2173	Deduction of abstraction subclass from implication.
|- (ph -> (ps -> x e. A))   =>   |- (ph -> {x | ps} (_ A)
Theorem	abssi 2174	Inference of abstraction subclass from implication.
|- (ph -> x e. A)   =>   |- {x | ph} (_ A
Theorem	ss2rab 2175	Restricted abstraction classes in a subclass relationship.
|- ({x e. A | ph} (_ {x e. A | ps} <-> A.x e. A (ph -> ps))
Theorem	rabss 2176	Restricted class abstraction in a subclass relationship.
|- ({x e. A | ph} (_ B <-> A.x e. A (ph -> x e. B))
Theorem	ssrab 2177	Subclass of a restricted class abstraction.
|- (B (_ {x e. A | ph} <-> (B (_ A /\ A.x e. B ph))
Theorem	ssrabdv 2178	Subclass of a restricted class abstraction (deduction rule).
|- (ph -> B (_ A)   &   |- ((ph /\ x e. B) -> ps)   =>   |- (ph -> B (_ {x e. A | ps})
Theorem	ss2rabdv 2179	Deduction of restricted abstraction subclass from implication.
|- ((ph /\ x e. A) -> (ps -> ch))   =>   |- (ph -> {x e. A | ps} (_ {x e. A | ch})
Theorem	ss2rabi 2180	Inference of restricted abstraction subclass from implication.
|- (x e. A -> (ph -> ps))   =>   |- {x e. A | ph} (_ {x e. A | ps}
Theorem	rabss2 2181	Subclass law for restricted abstraction.
|- (A (_ B -> {x e. A | ph} (_ {x e. B | ph})
Theorem	ssab2 2182	Subclass relation for the restriction of a class abstraction.
|- {x | (x e. A /\ ph)} (_ A
Theorem	ssrab2 2183	Subclass relation for a restricted class.
|- {x e. A | ph} (_ A
Theorem	uniiunlem 2184	A subset relationship useful for converting union to indexed union using dfiun2 2655 or dfiun2g 2654 and intersection to indexed intersection using dfiin2 2656.
|- (A.x e. A B e. D -> (A.x e. A B e. C <-> {y | E.x e. A y = B} (_ C))
Theorem	dfpss2 2185	Alternate definition of proper subclass.
|- (A (. B <-> (A (_ B /\ -. A = B))
Theorem	dfpss3 2186	Alternate definition of proper subclass.
|- (A (. B <-> (A (_ B /\ -. B (_ A))
Theorem	psseq1 2187	Equality theorem for proper subclass.
|- (A = B -> (A (. C <-> B (. C))
Theorem	psseq2 2188	Equality theorem for proper subclass.
|- (A = B -> (C (. A <-> C (. B))
Theorem	psseq1i 2189	An equality inference for the proper subclass relationship.
|- A = B   =>   |- (A (. C <-> B (. C)
Theorem	psseq2i 2190	An equality inference for the proper subclass relationship.
|- A = B   =>   |- (C (. A <-> C (. B)
Theorem	psseq12i 2191	An equality inference for the proper subclass relationship.
|- A = B   &   |- C = D   =>   |- (A (. C <-> B (. D)
Theorem	psseq1d 2192	An equality deduction for the proper subclass relationship.
|- (ph -> A = B)   =>   |- (ph -> (A (. C <-> B (. C))
Theorem	psseq2d 2193	An equality deduction for the proper subclass relationship.
|- (ph -> A = B)   =>   |- (ph -> (C (. A <-> C (. B))
Theorem	psseq12d 2194	An equality deduction for the proper subclass relationship.
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> (A (. C <-> B (. D))
Theorem	pssss 2195	A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23.
|- (A (. B -> A (_ B)
Theorem	pssssd 2196	Deduce subclass from proper subclass.
|- (ph -> A (. B)   =>   |- (ph -> A (_ B)
Theorem	sspss 2197	Subclass in terms of proper subclass.
|- (A (_ B <-> (A (. B \/ A = B))
Theorem	pssirr 2198	Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23.
|- -. A (. A
Theorem	pssn2lp 2199	Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23.
|- -. (A (. B /\ B (. A)
Theorem	sspsstri 2200	Two ways of stating trichotomy with respect to inclusion.
|- ((A (_ B \/ B (_ A) <-> (A (. B \/ A = B \/ B (. A))