mmtheorems17 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 1601-1700 - Page 17 of 123

Type	Label	Description
Statement
Theorem	syl6eleq 1601	A membership and equality inference.
|- (ph -> A e. B)   &   |- B = C   =>   |- (ph -> A e. C)
Theorem	syl6eleqr 1602	A membership and equality inference.
|- (ph -> A e. B)   &   |- C = B   =>   |- (ph -> A e. C)
Theorem	cleqf 1603	Establish equality between classes, 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	nelneq 1604	A way of showing two classes are not equal.
|- ((A e. C /\ -. B e. C) -> -. A = B)
Theorem	nelneq2 1605	A way of showing two classes are not equal.
|- ((A e. B /\ -. A e. C) -> -. B = C)
Theorem	hbxfr 1606	A utility lemma to transfer a bound-variable hypothesis builder into a definition.
|- A = B   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. A -> A.x y e. A)
Theorem	hblem 1607	Lemma for hbeq 1608 and hbel 1609.
Theorem	hbeq 1608	If x is effectively bound in A and B, it is effectively bound in A = B.
|- (y e. A -> A.x y e. A)   &   |- (z e. B -> A.x z e. B)   =>   |- (A = B -> A.x A = B)
Theorem	hbel 1609	If x is effectively bound in A and B, it is effectively bound in A e. B.
|- (y e. A -> A.x y e. A)   &   |- (z e. B -> A.x z e. B)   =>   |- (A e. B -> A.x A e. B)
Theorem	hbeleq 1610	If x is effectively bound in y e. A, then it is effectively bound in y = A.
|- (y e. A -> A.x y e. A)   =>   |- (y = A -> A.x y = A)
Theorem	abeq2 1611	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 1616 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 ph (that has a free variable x) to a theorem with a class variable A, we substitute x e. A for ph throughout and simplify, where A is a new class variable not already in the wff. An example is the conversion of zfauscl 2779 to inex1 2790 (look at the instance of zfauscl 2779 that occurs in the proof of inex1 2790). Conversely, to convert a theorem with a class variable A to one with ph, we substitute {x | ph} for A throughout and simplify, where x and ph are new set and wff variables not already in the wff. An example is cp 4868, which derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 4867.
|- (A = {x | ph} <-> A.x(x e. A <-> ph))
Theorem	abeq1 1612	Equality of a class variable and a class abstraction.
|- ({x | ph} = A <-> A.x(ph <-> x e. A))
Theorem	abeq2i 1613	Equality of a class variable and a class abstraction (inference rule).
|- A = {x | ph}   =>   |- (x e. A <-> ph)
Theorem	abeq1i 1614	Equality of a class variable and a class abstraction (inference rule).
|- {x | ph} = A   =>   |- (ph <-> x e. A)
Theorem	abeq2d 1615	Equality of a class variable and a class abstraction (deduction).
|- (ph -> A = {x | ps})   =>   |- (ph -> (x e. A <-> ps))
Theorem	eq2ab 1616	Equality of two class abstractions means their wff's are equivalent.
|- ({x | ph} = {x | ps} <-> A.x(ph <-> ps))
Theorem	abbi2i 1617	Equality of a class variable and a class abstraction (inference rule).
|- (x e. A <-> ph)   =>   |- A = {x | ph}
Theorem	abbii 1618	Equivalent wff's yield equal class abstractions (inference rule).
|- (ph <-> ps)   =>   |- {x | ph} = {x | ps}
Theorem	abbid 1619	Equivalent wff's yield equal class abstractions (deduction rule).
|- (ph -> A.xph)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> {x | ps} = {x | ch})
Theorem	abbidv 1620	Equivalent wff's yield equal class abstractions (deduction rule).
|- (ph -> (ps <-> ch))   =>   |- (ph -> {x | ps} = {x | ch})
Theorem	abbi2dv 1621	Deduction from a wff to a class abstraction.
|- (ph -> (x e. A <-> ps))   =>   |- (ph -> A = {x | ps})
Theorem	abbi1dv 1622	Deduction from a wff to a class abstraction.
|- (ph -> (ps <-> x e. A))   =>   |- (ph -> {x | ps} = A)
Theorem	abid2 1623	A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35.
|- {x | x e. A} = A
Theorem	clelab 1624	Membership of a class variable in a class abstraction.
|- (A e. {x | ph} <-> E.x(x = A /\ ph))
Theorem	clabel 1625	Membership of a class abstraction in another class
|- ({x | ph} e. A <-> E.y(y e. A /\ A.x(x e. y <-> ph)))
Theorem	sbab 1626	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 e. A})
Theorem	sbabel 1627	Theorem to move a substitution in and out of a class abstraction.
|- (w e. A -> A.x w e. A)   =>   |- ([y / x]{z | ph} e. A <-> {z | [y / x]ph} e. A)
Negated equality and membership
Syntax	wne 1628	Extend wff notation to include inequality.
wff A =/= B
Syntax	wnel 1629	Extend wff notation to include negated membership.
wff A e/ B
Definition	df-ne 1630	Define inequality.
|- (A =/= B <-> -. A = B)
Definition	df-nel 1631	Define negated membership.
|- (A e/ B <-> -. A e. B)
Theorem	nne 1632	Negation of inequality.
|- (-. A =/= B <-> A = B)
Theorem	neeq1 1633	Equality theorem for inequality.
|- (A = B -> (A =/= C <-> B =/= C))
Theorem	neeq2 1634	Equality theorem for inequality.
|- (A = B -> (C =/= A <-> C =/= B))
Theorem	neeq1i 1635	Inference for inequality.
|- A = B   =>   |- (A =/= C <-> B =/= C)
Theorem	neeq2i 1636	Inference for inequality.
|- A = B   =>   |- (C =/= A <-> C =/= B)
Theorem	neeq1d 1637	Deduction for inequality.
|- (ph -> A = B)   =>   |- (ph -> (A =/= C <-> B =/= C))
Theorem	neeq2d 1638	Deduction for inequality.
|- (ph -> A = B)   =>   |- (ph -> (C =/= A <-> C =/= B))
Theorem	necon3abii 1639	Deduction from equality to inequality.
|- (A = B <-> ph)   =>   |- (A =/= B <-> -. ph)
Theorem	necon3bbii 1640	Deduction from equality to inequality.
|- (ph <-> A = B)   =>   |- (-. ph <-> A =/= B)
Theorem	necon3bii 1641	Inference from equality to inequality.
|- (A = B <-> C = D)   =>   |- (A =/= B <-> C =/= D)
Theorem	necon3abid 1642	Deduction from equality to inequality.
|- (ph -> (A = B <-> ps))   =>   |- (ph -> (A =/= B <-> -. ps))
Theorem	necon3bbid 1643	Deduction from equality to inequality.
|- (ph -> (ps <-> A = B))   =>   |- (ph -> (-. ps <-> A =/= B))
Theorem	necon3bid 1644	Deduction from equality to inequality.
|- (ph -> (A = B <-> C = D))   =>   |- (ph -> (A =/= B <-> C =/= D))
Theorem	necon3ad 1645	Contrapositive law deduction for inequality.
|- (ph -> (ps -> A = B))   =>   |- (ph -> (A =/= B -> -. ps))
Theorem	necon3bd 1646	Contrapositive law deduction for inequality.
|- (ph -> (A = B -> ps))   =>   |- (ph -> (-. ps -> A =/= B))
Theorem	necon3d 1647	Contrapositive law deduction for inequality.
|- (ph -> (A = B -> C = D))   =>   |- (ph -> (C =/= D -> A =/= B))
Theorem	necon3i 1648	Contrapositive inference for inequality.
|- (A = B -> C = D)   =>   |- (C =/= D -> A =/= B)
Theorem	necon3ai 1649	Contrapositive inference for inequality.
|- (ph -> A = B)   =>   |- (A =/= B -> -. ph)
Theorem	necon3bi 1650	Contrapositive inference for inequality.
|- (A = B -> ph)   =>   |- (-. ph -> A =/= B)
Theorem	necon1ai 1651	Contrapositive inference for inequality.
|- (-. ph -> A = B)   =>   |- (A =/= B -> ph)
Theorem	necon1bi 1652	Contrapositive inference for inequality.
|- (A =/= B -> ph)   =>   |- (-. ph -> A = B)
Theorem	necon1i 1653	Contrapositive inference for inequality.
|- (A =/= B -> C = D)   =>   |- (C =/= D -> A = B)
Theorem	necon2ai 1654	Contrapositive inference for inequality.
|- (A = B -> -. ph)   =>   |- (ph -> A =/= B)
Theorem	necon2bi 1655	Contrapositive inference for inequality.
|- (ph -> A =/= B)   =>   |- (A = B -> -. ph)
Theorem	necon2i 1656	Contrapositive inference for inequality.
|- (A = B -> C =/= D)   =>   |- (C = D -> A =/= B)
Theorem	necon2ad 1657	Contrapositive inference for inequality.
|- (ph -> (A = B -> -. ps))   =>   |- (ph -> (ps -> A =/= B))
Theorem	necon2bd 1658	Contrapositive inference for inequality.
|- (ph -> (ps -> A =/= B))   =>   |- (ph -> (A = B -> -. ps))
Theorem	necon2d 1659	Contrapositive inference for inequality.
|- (ph -> (A = B -> C =/= D))   =>   |- (ph -> (C = D -> A =/= B))
Theorem	necon1abii 1660	Contrapositive inference for inequality.
|- (-. ph <-> A = B)   =>   |- (A =/= B <-> ph)
Theorem	necon1bbii 1661	Contrapositive inference for inequality.
|- (A =/= B <-> ph)   =>   |- (-. ph <-> A = B)
Theorem	necon1abid 1662	Contrapositive deduction for inequality.
|- (ph -> (-. ps <-> A = B))   =>   |- (ph -> (A =/= B <-> ps))
Theorem	necon1bbid 1663	Contrapositive inference for inequality.
|- (ph -> (A =/= B <-> ps))   =>   |- (ph -> (-. ps <-> A = B))
Theorem	necon2abii 1664	Contrapositive inference for inequality.
|- (A = B <-> -. ph)   =>   |- (ph <-> A =/= B)
Theorem	necon2bbii 1665	Contrapositive inference for inequality.
|- (ph <-> A =/= B)   =>   |- (A = B <-> -. ph)
Theorem	necon2abid 1666	Contrapositive deduction for inequality.
|- (ph -> (A = B <-> -. ps))   =>   |- (ph -> (ps <-> A =/= B))
Theorem	necon2bbid 1667	Contrapositive deduction for inequality.
|- (ph -> (ps <-> A =/= B))   =>   |- (ph -> (A = B <-> -. ps))
Theorem	necon4ai 1668	Contrapositive inference for inequality.
|- (A =/= B -> -. ph)   =>   |- (ph -> A = B)
Theorem	necon4i 1669	Contrapositive inference for inequality.
|- (A =/= B -> C =/= D)   =>   |- (C = D -> A = B)
Theorem	necon4ad 1670	Contrapositive inference for inequality.
|- (ph -> (A =/= B -> -. ps))   =>   |- (ph -> (ps -> A = B))
Theorem	necon4bd 1671	Contrapositive inference for inequality.
|- (ph -> (-. ps -> A =/= B))   =>   |- (ph -> (A = B -> ps))
Theorem	necon4d 1672	Contrapositive inference for inequality.
|- (ph -> (A =/= B -> C =/= D))   =>   |- (ph -> (C = D -> A = B))
Theorem	necon4abid 1673	Contrapositive law deduction for inequality.
|- (ph -> (A =/= B <-> -. ps))   =>   |- (ph -> (A = B <-> ps))
Theorem	necon4bid 1674	Contrapositive law deduction for inequality.
|- (ph -> (A =/= B <-> C =/= D))   =>   |- (ph -> (A = B <-> C = D))
Theorem	necon1ad 1675	Contrapositive deduction for inequality.
|- (ph -> (-. ps -> A = B))   =>   |- (ph -> (A =/= B -> ps))
Theorem	necon1bd 1676	Contrapositive deduction for inequality.
|- (ph -> (A =/= B -> ps))   =>   |- (ph -> (-. ps -> A = B))
Theorem	necon1d 1677	Contrapositive law deduction for inequality.
|- (ph -> (A =/= B -> C = D))   =>   |- (ph -> (C =/= D -> A = B))
Theorem	nebi 1678	Contraposition law for inequality.
|- ((A = B <-> C = D) <-> (A =/= B <-> C =/= D))
Theorem	pm2.61ne 1679	Deduction eliminating an inequality in an antecedent.
|- (A = B -> (ps <-> ch))   &   |- ((ph /\ A =/= B) -> ps)   &   |- (ph -> ch)   =>   |- (ph -> ps)
Theorem	pm2.61ine 1680	Inference eliminating an inequality in an antecedent.
|- (A = B -> ph)   &   |- (A =/= B -> ph)   =>   |- ph
Theorem	pm2.61dne 1681	Deduction eliminating an inequality in an antecedent.
|- (ph -> (A = B -> ps))   &   |- (ph -> (A =/= B -> ps))   =>   |- (ph -> ps)
Theorem	necom 1682	Commutation of inequality.
|- (A =/= B <-> B =/= A)
Theorem	necomd 1683	Deduction from commutative law for inequality.
|- (ph -> A =/= B)   =>   |- (ph -> B =/= A)
Theorem	neor 1684	Logical OR with an equality.
|- ((A = B \/ ps) <-> (A =/= B -> ps))
Theorem	neanior 1685	A DeMorgan's law for inequality.
|- ((A =/= B /\ C =/= D) <-> -. (A = B \/ C = D))
Theorem	neorian 1686	A DeMorgan's law for inequality.
|- ((A =/= B \/ C =/= D) <-> -. (A = B /\ C = D))
Theorem	nemtbir 1687	An inference from an inequality, related to modus tollens.
|- A =/= B   &   |- (ph <-> A = B)   =>   |- -. ph
Theorem	neleq1 1688	Equality theorem for negated membership.
|- (A = B -> (A e/ C <-> B e/ C))
Theorem	neleq2 1689	Equality theorem for negated membership.
|- (A = B -> (C e/ A <-> C e/ B))
Theorem	hbne 1690	Bound-variable hypothesis builder for inequality.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (A =/= B -> A.x A =/= B)
Restricted quantification
Syntax	wral 1691	Extend wff notation to include restricted universal quantification.
wff A.x e. A ph
Syntax	wrex 1692	Extend wff notation to include restricted existential quantification.
wff E.x e. A ph
Syntax	wreu 1693	Extend wff notation to include restricted existential uniqueness.
wff E!x e. A ph
Syntax	crab 1694	Extend class notation to include the restricted class abstraction (class builder).
class {x e. A | ph}
Definition	df-ral 1695	Define restricted universal quantification. Special case of Definition 4.15(3) of [TakeutiZaring] p. 22.
|- (A.x e. A ph <-> A.x(x e. A -> ph))
Definition	df-rex 1696	Define restricted existential quantification. Special case of Definition 4.15(4) of [TakeutiZaring] p. 22.
|- (E.x e. A ph <-> E.x(x e. A /\ ph))
Definition	df-reu 1697	Define restricted existential uniqueness.
|- (E!x e. A ph <-> E!x(x e. A /\ ph))
Definition	df-rab 1698	Define a restricted class abstraction (class builder), which is the class of all x in A such that ph is true. Definition of [TakeutiZaring] p. 20.
|- {x e. A | ph} = {x | (x e. A /\ ph)}
Theorem	ralnex 1699	Relationship between restricted universal and existential quantifiers.
|- (A.x e. A -. ph <-> -. E.x e. A ph)
Theorem	rexnal 1700	Relationship between restricted universal and existential quantifiers.
|- (E.x e. A -. ph <-> -. A.x e. A ph)