Metamath Proof Explorer - mmtheorems12

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Statement List for Metamath Proof Explorer - 1101-1200 - Page 12 of 58

Type	Label	Description
Statement
Theorem	cleqri 1101	Infer equality of sets from equivalence of membership.
⊢ (x ∈ A ↔ x ∈ B)    ⇒   ⊢ A = B
Theorem	cleqid 1102	Class identity law (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41.
⊢ A = A
Theorem	cleqcom 1103	Commutative law for class equality. Theorem 6.5 of [Quine] p. 41.
⊢ (A = B ↔ B = A)
Theorem	cleqcoms 1104	Inference applying commutative law for class equality to an antecedent.
⊢ (A = B → φ)    ⇒   ⊢ (B = A → φ)
Theorem	cleqcomi 1105	Inference from commutative law for class equality.
⊢ A = B    ⇒   ⊢ B = A
Theorem	cleqcomd 1106	Deduction from commutative law for class equality.
⊢ (φ → A = B)    ⇒   ⊢ (φ → B = A)
Theorem	cleq1 1107	Equality implies equivalence of equalities.
⊢ (A = B → (A = C ↔ B = C))
Theorem	cleq1i 1108	Inference from equality to equivalence of equalities.
⊢ A = B    ⇒   ⊢ (A = C ↔ B = C)
Theorem	cleq1d 1109	Deduction from equality to equivalence of equalities.
⊢ (φ → A = B)    ⇒   ⊢ (φ → (A = C ↔ B = C))
Theorem	cleq2 1110	Equality implies equivalence of equalities.
⊢ (A = B → (C = A ↔ C = B))
Theorem	cleq2i 1111	Inference from equality to equivalence of equalities.
⊢ A = B    ⇒   ⊢ (C = A ↔ C = B)
Theorem	cleq2d 1112	Deduction from equality to equivalence of equalities.
⊢ (φ → A = B)    ⇒   ⊢ (φ → (C = A ↔ C = B))
Theorem	cleq12 1113	Equality relationship among 4 classes.
⊢ ((A = B ∧ C = D) → (A = C ↔ B = D))
Theorem	cleq12i 1114	A useful inference for substituting definitions into an equality.
⊢ A = B    &   ⊢ C = D    ⇒   ⊢ (A = C ↔ B = D)
Theorem	cleq12d 1115	A useful inference for substituting definitions into an equality.
⊢ (φ → A = B)    &   ⊢ (φ → C = D)    ⇒   ⊢ (φ → (A = C ↔ B = D))
Theorem	cleqan12d 1116	A useful inference for substituting definitions into an equality.
⊢ (φ → A = B)    &   ⊢ (ψ → C = D)    ⇒   ⊢ ((φ ∧ ψ) → (A = C ↔ B = D))
Theorem	cleqan12rd 1117	A useful inference for substituting definitions into an equality.
⊢ (φ → A = B)    &   ⊢ (ψ → C = D)    ⇒   ⊢ ((ψ ∧ φ) → (A = C ↔ B = D))
Theorem	cleqtr 1118	Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13.
⊢ ((A = B ∧ B = C) → A = C)
Theorem	eqtr 1119	An equality transitivity inference.
⊢ A = B    &   ⊢ B = C    ⇒   ⊢ A = C
Theorem	eqtr2 1120	An equality transitivity inference.
⊢ A = B    &   ⊢ B = C    ⇒   ⊢ C = A
Theorem	eqtr3 1121	An equality transitivity inference.
⊢ A = B    &   ⊢ A = C    ⇒   ⊢ B = C
Theorem	eqtr4 1122	An equality transitivity inference.
⊢ A = B    &   ⊢ C = B    ⇒   ⊢ A = C
Theorem	3eqtr 1123	An inference from three chained equalities.
⊢ A = B    &   ⊢ B = C    &   ⊢ C = D    ⇒   ⊢ A = D
Theorem	3eqtr3 1124	An inference from three chained equalities.
⊢ A = B    &   ⊢ A = C    &   ⊢ B = D    ⇒   ⊢ C = D
Theorem	3eqtr3r 1125	An inference from three chained equalities.
⊢ A = B    &   ⊢ A = C    &   ⊢ B = D    ⇒   ⊢ D = C
Theorem	3eqtr4 1126	An inference from three chained equalities.
⊢ A = B    &   ⊢ C = A    &   ⊢ D = B    ⇒   ⊢ C = D
Theorem	3eqtr4r 1127	An inference from three chained equalities.
⊢ A = B    &   ⊢ C = A    &   ⊢ D = B    ⇒   ⊢ D = C
Theorem	eqtrd 1128	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ (φ → B = C)    ⇒   ⊢ (φ → A = C)
Theorem	eqtr2d 1129	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ (φ → B = C)    ⇒   ⊢ (φ → C = A)
Theorem	eqtr3d 1130	An equality transitivity equality deduction.
⊢ (φ → A = B)    &   ⊢ (φ → A = C)    ⇒   ⊢ (φ → B = C)
Theorem	eqtr4d 1131	An equality transitivity equality deduction.
⊢ (φ → A = B)    &   ⊢ (φ → C = B)    ⇒   ⊢ (φ → A = C)
Theorem	3eqtrd 1132	A deduction from three chained equalities.
⊢ (φ → A = B)    &   ⊢ (φ → B = C)    &   ⊢ (φ → C = D)    ⇒   ⊢ (φ → A = D)
Theorem	3eqtr3d 1133	A deduction from three chained equalities.
⊢ (φ → A = B)    &   ⊢ (φ → A = C)    &   ⊢ (φ → B = D)    ⇒   ⊢ (φ → C = D)
Theorem	3eqtr4d 1134	A deduction from three chained equalities.
⊢ (φ → A = B)    &   ⊢ (φ → C = A)    &   ⊢ (φ → D = B)    ⇒   ⊢ (φ → C = D)
Theorem	3eqtr4rd 1135	A deduction from three chained equalities.
⊢ (φ → A = B)    &   ⊢ (φ → C = A)    &   ⊢ (φ → D = B)    ⇒   ⊢ (φ → D = C)
Theorem	syl5eq 1136	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ C = A    ⇒   ⊢ (φ → C = B)
Theorem	syl5req 1137	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ C = A    ⇒   ⊢ (φ → B = C)
Theorem	syl5eqr 1138	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ A = C    ⇒   ⊢ (φ → C = B)
Theorem	syl5reqr 1139	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ A = C    ⇒   ⊢ (φ → B = C)
Theorem	syl6eq 1140	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ B = C    ⇒   ⊢ (φ → A = C)
Theorem	syl6req 1141	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ B = C    ⇒   ⊢ (φ → C = A)
Theorem	syl6eqr 1142	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ C = B    ⇒   ⊢ (φ → A = C)
Theorem	syl6reqr 1143	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ C = B    ⇒   ⊢ (φ → C = A)
Theorem	sylan9eq 1144	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ (ψ → B = C)    ⇒   ⊢ ((φ ∧ ψ) → A = C)
Theorem	sylan9eqr 1145	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ (ψ → B = C)    ⇒   ⊢ ((ψ ∧ φ) → A = C)
Theorem	3eqtr3g 1146	A chained equality inference, useful for converting from definitions.
⊢ (φ → A = B)    &   ⊢ A = C    &   ⊢ B = D    ⇒   ⊢ (φ → C = D)
Theorem	3eqtr4g 1147	A chained equality inference, useful for converting to definitions.
⊢ (φ → A = B)    &   ⊢ C = A    &   ⊢ D = B    ⇒   ⊢ (φ → C = D)
Theorem	cleq2tr 1148	A compound transitive inference for class equality.
⊢ (A = C → D = F)    &   ⊢ (B = D → C = G)    ⇒   ⊢ ((A = C ∧ B = F) ↔ (B = D ∧ A = G))
Theorem	eleq1 1149	Equality implies equivalence of membership.
⊢ (A = B → (A ∈ C ↔ B ∈ C))
Theorem	eleq2 1150	Equality implies equivalence of membership.
⊢ (A = B → (C ∈ A ↔ C ∈ B))
Theorem	eleq12 1151	Equality implies equivalence of membership.
⊢ ((A = B ∧ C = D) → (A ∈ C ↔ B ∈ D))
Theorem	eleq1i 1152	Inference from equality to equivalence of membership.
⊢ A = B    ⇒   ⊢ (A ∈ C ↔ B ∈ C)
Theorem	eleq2i 1153	Inference from equality to equivalence of membership.
⊢ A = B    ⇒   ⊢ (C ∈ A ↔ C ∈ B)
Theorem	eleq12i 1154	Inference from equality to equivalence of membership.
⊢ A = B    &   ⊢ C = D    ⇒   ⊢ (A ∈ C ↔ B ∈ D)
Theorem	eleq1d 1155	Deduction from equality to equivalence of membership.
⊢ (φ → A = B)    ⇒   ⊢ (φ → (A ∈ C ↔ B ∈ C))
Theorem	eleq2d 1156	Deduction from equality to equivalence of membership.
⊢ (φ → A = B)    ⇒   ⊢ (φ → (C ∈ A ↔ C ∈ B))
Theorem	eleq12d 1157	Deduction from equality to equivalence of membership.
⊢ (φ → A = B)    &   ⊢ (φ → C = D)    ⇒   ⊢ (φ → (A ∈ C ↔ B ∈ D))
Theorem	eleq1a 1158	A transitive-type law relating membership and equality.
⊢ (A ∈ B → (C = A → C ∈ B))
Theorem	eqeltr 1159	Substitution of equal classes into membership relation.
⊢ A = B    &   ⊢ B ∈ C    ⇒   ⊢ A ∈ C
Theorem	eqeltrr 1160	Substitution of equal classes into membership relation.
⊢ A = B    &   ⊢ A ∈ C    ⇒   ⊢ B ∈ C
Theorem	eleqtr 1161	Substitution of equal classes into membership relation.
⊢ A ∈ B    &   ⊢ B = C    ⇒   ⊢ A ∈ C
Theorem	eleqtrr 1162	Substitution of equal classes into membership relation.
⊢ A ∈ B    &   ⊢ C = B    ⇒   ⊢ A ∈ C
Theorem	eqeltrd 1163	Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-02.)
⊢ (φ → A = B)    &   ⊢ (φ → B ∈ C)    ⇒   ⊢ (φ → A ∈ C)
Theorem	eqeltrrd 1164	Deduction that substitutes equal classes into membership.
⊢ (φ → A = B)    &   ⊢ (φ → A ∈ C)    ⇒   ⊢ (φ → B ∈ C)
Theorem	eleqtrd 1165	Deduction that substitutes equal classes into membership.
⊢ (φ → A ∈ B)    &   ⊢ (φ → B = C)    ⇒   ⊢ (φ → A ∈ C)
Theorem	eleqtrrd 1166	Deduction that substitutes equal classes into membership.
⊢ (φ → A ∈ B)    &   ⊢ (φ → C = B)    ⇒   ⊢ (φ → A ∈ C)
Theorem	cleqf 1167	Establish equality between classes, requiring only that x not be 'free' in A and B (but not necessarily absent from them).
⊢ (y ∈ A → ∀x y ∈ A)    &   ⊢ (y ∈ B → ∀x y ∈ B)    ⇒   ⊢ (A = B ↔ ∀x(x ∈ A ↔ x ∈ B))
Theorem	clneq 1168	A way of showing two classes are not equal.
⊢ ((A ∈ C ∧ ¬ B ∈ C) → ¬ A = B)
Theorem	clneq2 1169	A way of showing two classes are not equal.
⊢ ((A ∈ B ∧ ¬ A ∈ C) → ¬ B = C)
Theorem	hblem 1170	Lemma for hbeq 1171 and hbel 1172.
⊢ (y ∈ A → ∀x y ∈ A)    ⇒   ⊢ (z ∈ A → ∀x z ∈ A)
Theorem	hbeq 1171	If x is effectively bound in A and B, it is effectively bound in A = B.
⊢ (y ∈ A → ∀x y ∈ A)    &   ⊢ (z ∈ B → ∀x z ∈ B)    ⇒   ⊢ (A = B → ∀x A = B)
Theorem	hbel 1172	If x is effectively bound in A and B, it is effectively bound in A ∈ B.
⊢ (y ∈ A → ∀x y ∈ A)    &   ⊢ (z ∈ B → ∀x z ∈ B)    ⇒   ⊢ (A ∈ B → ∀x A ∈ B)
Theorem	hbeleq 1173	If x is effectively bound in y ∈ A, then it is effectively bound in y = A.
⊢ (y ∈ A → ∀x y ∈ A)    ⇒   ⊢ (y = A → ∀x y = A)
Theorem	cleqab 1174	Equality of a class variable and an class abstraction. Theorem 5.1 of [Quine] p. 34. This theorem is useful for converting expressions with class abstractions to expressions with class variables. Note that cleq2ab 1179 and its relatives are among those useful for converting theorems with class variables to equivalent theorems with wff variables, by first substituting an class abstraction for each class variable. Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable φ (that has a free variable x) to a theorem with a class variable A, we substitute x ∈ A for φ throughout and simplify, where A is a new class variable not already in the wff. An example is the conversion of zfaus 1480 to inex1 1697 (look at the instance of zfaus 1480 that occurs in the proof of inex1 1697). Conversely, to convert a theorem with a class variable A to one with φ, we substitute {x∣φ} for A throughout and simplify. An example is cp 3547, which derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 3546.
⊢ (A = {x∣φ} ↔ ∀x(x ∈ A ↔ φ))
Theorem	cleqabr 1175	Equality of a class variable and a class abstraction.
⊢ ({x∣φ} = A ↔ ∀x(φ ↔ x ∈ A))
Theorem	cleqabi 1176	Equality of a class variable and a class abstraction (inference rule).
⊢ A = {x∣φ}    ⇒   ⊢ (x ∈ A ↔ φ)
Theorem	cleqabri 1177	Equality of a class variable and a class abstraction (inference rule).
⊢ {x∣φ} = A    ⇒   ⊢ (φ ↔ x ∈ A)
Theorem	cleqabd 1178	Equality of a class variable and a class abstraction (deduction).
⊢ (φ → A = {x∣ψ})    ⇒   ⊢ (φ → (x ∈ A ↔ ψ))
Theorem	cleq2ab 1179	Equality of two class abstractions means their wff's are equivalent.
⊢ ({x∣φ} = {x∣ψ} ↔ ∀x(φ ↔ ψ))
Theorem	biabri 1180	Equality of a class variable and a class abstraction (inference rule).
⊢ (x ∈ A ↔ φ)    ⇒   ⊢ A = {x∣φ}
Theorem	biabi 1181	Equivalent wff's yield equal class abstractions (inference rule).
⊢ (φ ↔ ψ)    ⇒   ⊢ {x∣φ} = {x∣ψ}
Theorem	biabd 1182	Equivalent wff's yield equal class abstractions (deduction rule).
⊢ (φ → ∀xφ)    &   ⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → {x∣ψ} = {x∣χ})
Theorem	biabdv 1183	Equivalent wff's yield equal class abstractions (deduction rule).
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → {x∣ψ} = {x∣χ})
Theorem	biabrdv 1184	Deduction from a wff to a class abstraction.
⊢ (φ → (x ∈ A ↔ ψ))    ⇒   ⊢ (φ → A = {x∣ψ})
Theorem	biabldv 1185	Deduction from a wff to a class abstraction.
⊢ (φ → (ψ ↔ x ∈ A))    ⇒   ⊢ (φ → {x∣ψ} = A)
Theorem	abid2 1186	A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35.
⊢ {x∣x ∈ A} = A
Theorem	clelab 1187	Membership of a class variable in a class abstraction.
⊢ (A ∈ {x∣φ} ↔ ∃x(x = A ∧ φ))
Theorem	sbab 1188	The right-hand side of the second equality is a way of representing proper substitution of y for x into a class variable.
⊢ (x = y → A = {z∣[y / x]z ∈ A})
Theorem	sbabel 1189	Theorem to move a substitution in and out of a class abstraction.
⊢ (w ∈ A → ∀x w ∈ A)    ⇒   ⊢ ([y / x]{z∣φ} ∈ A ↔ {z∣[y / x]φ} ∈ A)
Syntax	wne 1190	Extend wff notation to include inequality.
wff A ≠ B
Syntax	wnel 1191	Extend wff notation to include negated membership.
wff A ∉ B
Definition	df-ne 1192	Define inequality.
⊢ (A ≠ B ↔ ¬ A = B)
Definition	df-nel 1193	Define negated membership.
⊢ (A ∉ B ↔ ¬ A ∈ B)
Theorem	neeq1 1194	Equality theorem for inequality.
⊢ (A = B → (A ≠ C ↔ B ≠ C))
Theorem	neeq2 1195	Equality theorem for inequality.
⊢ (A = B → (C ≠ A ↔ C ≠ B))
Theorem	neeq1d 1196	Deduction for inequality.
⊢ (φ → A = B)    ⇒   ⊢ (φ → (A ≠ C ↔ B ≠ C))
Theorem	neeq2d 1197	Deduction for inequality.
⊢ (φ → A = B)    ⇒   ⊢ (φ → (C ≠ A ↔ C ≠ B))
Theorem	necom 1198	Commutation of inequality.
⊢ (A ≠ B ↔ B ≠ A)
Theorem	neleq1 1199	Equality theorem for negated membership.
⊢ (A = B → (A ∉ C ↔ B ∉ C))
Theorem	neleq2 1200	Equality theorem for negated membership.
⊢ (A = B → (C ∉ A ↔ C ∉ B))