mmtheorems16 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 1501-1600 - Page 16 of 107

Type	Label	Description
Statement
Theorem	3eqtr3 1501	An inference from three chained equalities.
⊢ A = B    &   ⊢ A = C    &   ⊢ B = D    ⇒   ⊢ C = D
Theorem	3eqtr3r 1502	An inference from three chained equalities.
⊢ A = B    &   ⊢ A = C    &   ⊢ B = D    ⇒   ⊢ D = C
Theorem	3eqtr4 1503	An inference from three chained equalities.
⊢ A = B    &   ⊢ C = A    &   ⊢ D = B    ⇒   ⊢ C = D
Theorem	3eqtr4r 1504	An inference from three chained equalities.
⊢ A = B    &   ⊢ C = A    &   ⊢ D = B    ⇒   ⊢ D = C
Theorem	eqtrd 1505	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ (φ → B = C)    ⇒   ⊢ (φ → A = C)
Theorem	eqtr2d 1506	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ (φ → B = C)    ⇒   ⊢ (φ → C = A)
Theorem	eqtr3d 1507	An equality transitivity equality deduction.
⊢ (φ → A = B)    &   ⊢ (φ → A = C)    ⇒   ⊢ (φ → B = C)
Theorem	eqtr4d 1508	An equality transitivity equality deduction.
⊢ (φ → A = B)    &   ⊢ (φ → C = B)    ⇒   ⊢ (φ → A = C)
Theorem	3eqtrd 1509	A deduction from three chained equalities.
⊢ (φ → A = B)    &   ⊢ (φ → B = C)    &   ⊢ (φ → C = D)    ⇒   ⊢ (φ → A = D)
Theorem	3eqtrrd 1510	A deduction from three chained equalities.
⊢ (φ → A = B)    &   ⊢ (φ → B = C)    &   ⊢ (φ → C = D)    ⇒   ⊢ (φ → D = A)
Theorem	3eqtr2d 1511	A deduction from three chained equalities.
⊢ (φ → A = B)    &   ⊢ (φ → C = B)    &   ⊢ (φ → C = D)    ⇒   ⊢ (φ → A = D)
Theorem	3eqtr2rd 1512	A deduction from three chained equalities.
⊢ (φ → A = B)    &   ⊢ (φ → C = B)    &   ⊢ (φ → C = D)    ⇒   ⊢ (φ → D = A)
Theorem	3eqtr3d 1513	A deduction from three chained equalities.
⊢ (φ → A = B)    &   ⊢ (φ → A = C)    &   ⊢ (φ → B = D)    ⇒   ⊢ (φ → C = D)
Theorem	3eqtr3rd 1514	A deduction from three chained equalities.
⊢ (φ → A = B)    &   ⊢ (φ → A = C)    &   ⊢ (φ → B = D)    ⇒   ⊢ (φ → D = C)
Theorem	3eqtr4d 1515	A deduction from three chained equalities.
⊢ (φ → A = B)    &   ⊢ (φ → C = A)    &   ⊢ (φ → D = B)    ⇒   ⊢ (φ → C = D)
Theorem	3eqtr4rd 1516	A deduction from three chained equalities.
⊢ (φ → A = B)    &   ⊢ (φ → C = A)    &   ⊢ (φ → D = B)    ⇒   ⊢ (φ → D = C)
Theorem	syl5eq 1517	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ C = A    ⇒   ⊢ (φ → C = B)
Theorem	syl5req 1518	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ C = A    ⇒   ⊢ (φ → B = C)
Theorem	syl5eqr 1519	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ A = C    ⇒   ⊢ (φ → C = B)
Theorem	syl5reqr 1520	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ A = C    ⇒   ⊢ (φ → B = C)
Theorem	syl6eq 1521	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ B = C    ⇒   ⊢ (φ → A = C)
Theorem	syl6req 1522	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ B = C    ⇒   ⊢ (φ → C = A)
Theorem	syl6eqr 1523	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ C = B    ⇒   ⊢ (φ → A = C)
Theorem	syl6reqr 1524	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ C = B    ⇒   ⊢ (φ → C = A)
Theorem	sylan9eq 1525	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ (ψ → B = C)    ⇒   ⊢ ((φ ⋀ ψ) → A = C)
Theorem	sylan9req 1526	An equality transitivity deduction.
⊢ (φ → B = A)    &   ⊢ (ψ → B = C)    ⇒   ⊢ ((φ ⋀ ψ) → A = C)
Theorem	sylan9eqr 1527	An equality transitivity deduction.
⊢ (φ → A = B)    &   ⊢ (ψ → B = C)    ⇒   ⊢ ((ψ ⋀ φ) → A = C)
Theorem	3eqtr3g 1528	A chained equality inference, useful for converting from definitions.
⊢ (φ → A = B)    &   ⊢ A = C    &   ⊢ B = D    ⇒   ⊢ (φ → C = D)
Theorem	3eqtr4g 1529	A chained equality inference, useful for converting to definitions.
⊢ (φ → A = B)    &   ⊢ C = A    &   ⊢ D = B    ⇒   ⊢ (φ → C = D)
Theorem	3eqtr4a 1530	A chained equality inference, useful for converting to definitions.
⊢ A = B    &   ⊢ (φ → C = A)    &   ⊢ (φ → D = B)    ⇒   ⊢ (φ → C = D)
Theorem	eq2tr 1531	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 1532	Equality implies equivalence of membership.
⊢ (A = B → (A ∈ C ↔ B ∈ C))
Theorem	eleq2 1533	Equality implies equivalence of membership.
⊢ (A = B → (C ∈ A ↔ C ∈ B))
Theorem	eleq12 1534	Equality implies equivalence of membership.
⊢ ((A = B ⋀ C = D) → (A ∈ C ↔ B ∈ D))
Theorem	eleq1i 1535	Inference from equality to equivalence of membership.
⊢ A = B    ⇒   ⊢ (A ∈ C ↔ B ∈ C)
Theorem	eleq2i 1536	Inference from equality to equivalence of membership.
⊢ A = B    ⇒   ⊢ (C ∈ A ↔ C ∈ B)
Theorem	eleq12i 1537	Inference from equality to equivalence of membership.
⊢ A = B    &   ⊢ C = D    ⇒   ⊢ (A ∈ C ↔ B ∈ D)
Theorem	eleq1d 1538	Deduction from equality to equivalence of membership.
⊢ (φ → A = B)    ⇒   ⊢ (φ → (A ∈ C ↔ B ∈ C))
Theorem	eleq2d 1539	Deduction from equality to equivalence of membership.
⊢ (φ → A = B)    ⇒   ⊢ (φ → (C ∈ A ↔ C ∈ B))
Theorem	eleq12d 1540	Deduction from equality to equivalence of membership.
⊢ (φ → A = B)    &   ⊢ (φ → C = D)    ⇒   ⊢ (φ → (A ∈ C ↔ B ∈ D))
Theorem	eleq1a 1541	A transitive-type law relating membership and equality.
⊢ (A ∈ B → (C = A → C ∈ B))
Theorem	eqeltr 1542	Substitution of equal classes into membership relation.
⊢ A = B    &   ⊢ B ∈ C    ⇒   ⊢ A ∈ C
Theorem	eqeltrr 1543	Substitution of equal classes into membership relation.
⊢ A = B    &   ⊢ A ∈ C    ⇒   ⊢ B ∈ C
Theorem	eleqtr 1544	Substitution of equal classes into membership relation.
⊢ A ∈ B    &   ⊢ B = C    ⇒   ⊢ A ∈ C
Theorem	eleqtrr 1545	Substitution of equal classes into membership relation.
⊢ A ∈ B    &   ⊢ C = B    ⇒   ⊢ A ∈ C
Theorem	eqeltrd 1546	Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ (φ → A = B)    &   ⊢ (φ → B ∈ C)    ⇒   ⊢ (φ → A ∈ C)
Theorem	eqeltrrd 1547	Deduction that substitutes equal classes into membership.
⊢ (φ → A = B)    &   ⊢ (φ → A ∈ C)    ⇒   ⊢ (φ → B ∈ C)
Theorem	eleqtrd 1548	Deduction that substitutes equal classes into membership.
⊢ (φ → A ∈ B)    &   ⊢ (φ → B = C)    ⇒   ⊢ (φ → A ∈ C)
Theorem	eleqtrrd 1549	Deduction that substitutes equal classes into membership.
⊢ (φ → A ∈ B)    &   ⊢ (φ → C = B)    ⇒   ⊢ (φ → A ∈ C)
Theorem	syl5eqel 1550	A membership and equality inference.
⊢ (φ → A ∈ B)    &   ⊢ C = A    ⇒   ⊢ (φ → C ∈ B)
Theorem	syl5eqelr 1551	A membership and equality inference.
⊢ (φ → A ∈ B)    &   ⊢ A = C    ⇒   ⊢ (φ → C ∈ B)
Theorem	syl5eleq 1552	A membership and equality inference.
⊢ (φ → A = B)    &   ⊢ C ∈ A    ⇒   ⊢ (φ → C ∈ B)
Theorem	syl5eleqr 1553	A membership and equality inference.
⊢ (φ → B = A)    &   ⊢ C ∈ A    ⇒   ⊢ (φ → C ∈ B)
Theorem	syl6eqel 1554	A membership and equality inference.
⊢ (φ → A = B)    &   ⊢ B ∈ C    ⇒   ⊢ (φ → A ∈ C)
Theorem	syl6eqelr 1555	A membership and equality inference.
⊢ (φ → B = A)    &   ⊢ B ∈ C    ⇒   ⊢ (φ → A ∈ C)
Theorem	syl6eleq 1556	A membership and equality inference.
⊢ (φ → A ∈ B)    &   ⊢ B = C    ⇒   ⊢ (φ → A ∈ C)
Theorem	syl6eleqr 1557	A membership and equality inference.
⊢ (φ → A ∈ B)    &   ⊢ C = B    ⇒   ⊢ (φ → A ∈ C)
Theorem	cleqf 1558	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions.
⊢ (y ∈ A → ∀x y ∈ A)    &   ⊢ (y ∈ B → ∀x y ∈ B)    ⇒   ⊢ (A = B ↔ ∀x(x ∈ A ↔ x ∈ B))
Theorem	nelneq 1559	A way of showing two classes are not equal.
⊢ ((A ∈ C ⋀ ¬ B ∈ C) → ¬ A = B)
Theorem	nelneq2 1560	A way of showing two classes are not equal.
⊢ ((A ∈ B ⋀ ¬ A ∈ C) → ¬ B = C)
Theorem	hbxfr 1561	A utility lemma to transfer a bound-variable hypothesis builder into a definition.
⊢ A = B    &   ⊢ (y ∈ B → ∀x y ∈ B)    ⇒   ⊢ (y ∈ A → ∀x y ∈ A)
Theorem	hblem 1562	Lemma for hbeq 1563 and hbel 1564.
Theorem	hbeq 1563	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 1564	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 1565	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	abeq2 1566	Equality of a class variable and a class abstraction (also called a class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the relationship between expressions with class abstractions and expressions with class variables. Note that eq2ab 1571 and its relatives are among those useful for converting theorems with class variables to equivalent theorems with wff variables, by first substituting a 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 zfauscl 2701 to inex1 2712 (look at the instance of zfauscl 2701 that occurs in the proof of inex1 2712). Conversely, to convert a theorem with a class variable A to one with φ, we substitute {x∣φ} for A throughout and simplify, where x and φ are new set and wff variables not already in the wff. An example is cp 4705, which derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 4704.
⊢ (A = {x∣φ} ↔ ∀x(x ∈ A ↔ φ))
Theorem	abeq1 1567	Equality of a class variable and a class abstraction.
⊢ ({x∣φ} = A ↔ ∀x(φ ↔ x ∈ A))
Theorem	abeq2i 1568	Equality of a class variable and a class abstraction (inference rule).
⊢ A = {x∣φ}    ⇒   ⊢ (x ∈ A ↔ φ)
Theorem	abeq1i 1569	Equality of a class variable and a class abstraction (inference rule).
⊢ {x∣φ} = A    ⇒   ⊢ (φ ↔ x ∈ A)
Theorem	abeq2d 1570	Equality of a class variable and a class abstraction (deduction).
⊢ (φ → A = {x∣ψ})    ⇒   ⊢ (φ → (x ∈ A ↔ ψ))
Theorem	eq2ab 1571	Equality of two class abstractions means their wff's are equivalent.
⊢ ({x∣φ} = {x∣ψ} ↔ ∀x(φ ↔ ψ))
Theorem	abbi2i 1572	Equality of a class variable and a class abstraction (inference rule).
⊢ (x ∈ A ↔ φ)    ⇒   ⊢ A = {x∣φ}
Theorem	abbii 1573	Equivalent wff's yield equal class abstractions (inference rule).
⊢ (φ ↔ ψ)    ⇒   ⊢ {x∣φ} = {x∣ψ}
Theorem	abbid 1574	Equivalent wff's yield equal class abstractions (deduction rule).
⊢ (φ → ∀xφ)    &   ⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → {x∣ψ} = {x∣χ})
Theorem	abbidv 1575	Equivalent wff's yield equal class abstractions (deduction rule).
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → {x∣ψ} = {x∣χ})
Theorem	abbi2dv 1576	Deduction from a wff to a class abstraction.
⊢ (φ → (x ∈ A ↔ ψ))    ⇒   ⊢ (φ → A = {x∣ψ})
Theorem	abbi1dv 1577	Deduction from a wff to a class abstraction.
⊢ (φ → (ψ ↔ x ∈ A))    ⇒   ⊢ (φ → {x∣ψ} = A)
Theorem	abid2 1578	A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35.
⊢ {x∣x ∈ A} = A
Theorem	clelab 1579	Membership of a class variable in a class abstraction.
⊢ (A ∈ {x∣φ} ↔ ∃x(x = A ⋀ φ))
Theorem	clabel 1580	Membership of a class abstraction in another class
⊢ ({x∣φ} ∈ A ↔ ∃y(y ∈ A ⋀ ∀x(x ∈ y ↔ φ)))
Theorem	sbab 1581	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 1582	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)
Negated equality and membership
Syntax	wne 1583	Extend wff notation to include inequality.
wff A ≠ B
Syntax	wnel 1584	Extend wff notation to include negated membership.
wff A ∉ B
Definition	df-ne 1585	Define inequality.
⊢ (A ≠ B ↔ ¬ A = B)
Definition	df-nel 1586	Define negated membership.
⊢ (A ∉ B ↔ ¬ A ∈ B)
Theorem	nne 1587	Negation of inequality.
⊢ (¬ A ≠ B ↔ A = B)
Theorem	neeq1 1588	Equality theorem for inequality.
⊢ (A = B → (A ≠ C ↔ B ≠ C))
Theorem	neeq2 1589	Equality theorem for inequality.
⊢ (A = B → (C ≠ A ↔ C ≠ B))
Theorem	neeq1i 1590	Inference for inequality.
⊢ A = B    ⇒   ⊢ (A ≠ C ↔ B ≠ C)
Theorem	neeq2i 1591	Inference for inequality.
⊢ A = B    ⇒   ⊢ (C ≠ A ↔ C ≠ B)
Theorem	neeq1d 1592	Deduction for inequality.
⊢ (φ → A = B)    ⇒   ⊢ (φ → (A ≠ C ↔ B ≠ C))
Theorem	neeq2d 1593	Deduction for inequality.
⊢ (φ → A = B)    ⇒   ⊢ (φ → (C ≠ A ↔ C ≠ B))
Theorem	necon3abii 1594	Deduction from equality to inequality.
⊢ (A = B ↔ φ)    ⇒   ⊢ (A ≠ B ↔ ¬ φ)
Theorem	necon3bbii 1595	Deduction from equality to inequality.
⊢ (φ ↔ A = B)    ⇒   ⊢ (¬ φ ↔ A ≠ B)
Theorem	necon3bii 1596	Inference from equality to inequality.
⊢ (A = B ↔ C = D)    ⇒   ⊢ (A ≠ B ↔ C ≠ D)
Theorem	necon3abid 1597	Deduction from equality to inequality.
⊢ (φ → (A = B ↔ ψ))    ⇒   ⊢ (φ → (A ≠ B ↔ ¬ ψ))
Theorem	necon3bbid 1598	Deduction from equality to inequality.
⊢ (φ → (ψ ↔ A = B))    ⇒   ⊢ (φ → (¬ ψ ↔ A ≠ B))
Theorem	necon3bid 1599	Deduction from equality to inequality.
⊢ (φ → (A = B ↔ C = D))    ⇒   ⊢ (φ → (A ≠ B ↔ C ≠ D))
Theorem	necon3ad 1600	Contrapositive law deduction for inequality.
⊢ (φ → (ψ → A = B))    ⇒   ⊢ (φ → (A ≠ B → ¬ ψ))