mmtheorems16 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 1501-1600 - Page 16 of 123

Type	Label	Description
Statement
Theorem	axext2 1501	The Axiom of Extensionality (ax-ext 1500) restated so that it postulates the existence of a set z given two arbitrary sets x and y. This way to express it follows the general idea of the other ZFC axioms, which is to postulate the existence of sets given other sets.
|- E.z((z e. x <-> z e. y) -> x = y)
Theorem	axext3 1502	A generalization of the Axiom of Extensionality in which x and y need not be distinct.
|- (A.z(z e. x <-> z e. y) -> x = y)
Theorem	axext4 1503	A bidirectional version of Extensionality. Although this theorem "looks" like it is just a definition of equality, it requires the Axiom of Extensionality for its proof under our axiomatization. See the comments for ax-ext 1500 and df-cleq 1511.
|- (x = y <-> A.z(z e. x <-> z e. y))
Theorem	bm1.1 1504	Any set defined by a property is the only set defined by that property. Theorem 1.1 of [BellMachover] p. 462.
|- (ph -> A.xph)   =>   |- (E.xA.y(y e. x <-> ph) -> E!xA.y(y e. x <-> ph))
Class abstractions (a.k.a. class builders)
Syntax	cab 1505	Introduce the class builder or class abstraction notation ("the class of sets x such that ph is true"). Our class variables A, B, etc. range over class builders (implicitly in the case of defined class terms such as df-nul 2333). Note that a set variable can be expressed as a class builder per theorem cvjust 1513, justifying the assignment of set variables to class variables via the use of cv 991.
class {x | ph}
Definition	df-clab 1506	Define class abstraction notation (so-called by Quine), also called a "class builder" in the literature. x and y need not be distinct. Definition 2.1 of [Quine] p. 16. Typically, ph will have y as a free variable, and "{y | ph}" is read "the class of all sets y such that ph(y) is true." We do not define {y | ph} in isolation but only as part of an expression that extends or "overloads" the e. relationship. This is our first use of the e. symbol to connect classes instead of sets. The syntax definition wcel 994, which extends or "overloads" the wel 995 definition connecting set variables, requires that both sides of e. be a class. In df-cleq 1511 and df-clel 1514, we introduce a new kind of variable (class variable) that can substituted with expressions such as {y | ph}. In the present definition, the x on the left-hand side is a set variable. Syntax definition cv 991 allows us to substitute a set variable x for a class variable: all sets are classes by cvjust 1513 (but not necessarily vice-versa). For a full description of how classes are introduced and how to recover the primitive language, see the discussion in Quine (and under abeq2 1611 for a quick overview). Because class variables can be substituted with compound expressions and set variables cannot, it is often useful to convert a theorem containing a free set variable to a more general version with a class variable. This is done with theorems such as vtoclg 1893 which is used, for example, to convert elirrv 4741 to elirr 4742.
|- (x e. {y | ph} <-> [x / y]ph)
Theorem	abid 1507	Simplification of class abstraction notation when the free and bound variables are identical.
|- (x e. {x | ph} <-> ph)
Theorem	hbab1 1508	Bound-variable hypothesis builder for a class abstraction.
|- (y e. {x | ph} -> A.x y e. {x | ph})
Theorem	hbab 1509	Bound-variable hypothesis builder for a class abstraction.
|- (ph -> A.xph)   =>   |- (z e. {y | ph} -> A.x z e. {y | ph})
Theorem	hbabd 1510	Deduction form of bound-variable hypothesis builder hbab 1509.
|- (ph -> A.xA.yph)   &   |- (ph -> (ps -> A.xps))   =>   |- (ph -> (z e. {y | ps} -> A.x z e. {y | ps}))
Definition	df-cleq 1511	Define the equality connective between classes. Definition 2.7 of [Quine] p. 18. Also Definition 4.5 of [TakeutiZaring] p. 13; Chapter 4 provides its justification and methods for eliminating it. Note that its elimination will not necessarily result in a single wff in the original language but possibly a "scheme" of wffs. This is an example of a somewhat "risky" definition, meaning that it has a more complex than usual soundness justification (outside of Metamath), because it "overloads" or reuses the existing equality symbol rather than introducing a new symbol. This allows us to make statements that may not hold for the original symbol. For example, it permits us to deduce y = z <-> A.x(x e. y <-> x e. z), which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see theorem axext4 1503). We therefore include this axiom as a hypothesis, so that the use of Extensionality is properly indicated. We could avoid this complication by introducing a new symbol, say =2, in place of =. This would also have the advantage of making elimination of the definition straightforward, so that we could eliminate Extensionality as a hypothesis. We would then also have the advantage of being able to identify in various proofs exactly where Extensionality truly comes into play rather than just being an artifact of a definition.. One of our theorems would then be x =2 y <-> x = y by invoking Extensionality. However, to conform to literature usage, we retain this overloaded definition. This also makes some proofs shorter and probably easier to read, without the constant switching between two kinds of equality. See also comments under df-clab 1506, df-clel 1514, and abeq2 1611.
|- (A.x(x e. y <-> x e. z) -> y = z)   =>   |- (A = B <-> A.x(x e. A <-> x e. B))
Theorem	dfcleq 1512	The same as df-cleq 1511 with the hypothesis removed using the Axiom of Extensionality ax-ext 1500.
|- (A = B <-> A.x(x e. A <-> x e. B))
Theorem	cvjust 1513	Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a set variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 991, which allows us to substitute a set variable for a class variable. See also cab 1505 and df-clab 1506. Note that this is not a rigorous justification, because cv 991 is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class."
|- x = {y | y e. x}
Definition	df-clel 1514	Define the membership connective between classes. Theorem 6.3 of [Quine] p. 41, or Proposition 4.6 of [TakeutiZaring] p. 13, which we adopt as a definition. See these references for its metalogical justification. Note that like df-cleq 1511 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 1511 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 1366), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 1506.
|- (A e. B <-> E.x(x = A /\ x e. B))
Theorem	eqriv 1515	Infer equality of classes from equivalence of membership.
|- (x e. A <-> x e. B)   =>   |- A = B
Theorem	eqrdv 1516	Deduce equality of classes from equivalence of membership.
|- (ph -> (x e. A <-> x e. B))   =>   |- (ph -> A = B)
Theorem	eqrdav 1517	Deduce equality of classes from an equivalence of membership that depends on the membership variable.
|- ((ph /\ x e. A) -> x e. C)   &   |- ((ph /\ x e. B) -> x e. C)   &   |- ((ph /\ x e. C) -> (x e. A <-> x e. B))   =>   |- (ph -> A = B)
Theorem	eqid 1518	Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41. This law is thought to have originated with Aristotle (Metaphysics, Book VII, Part 17). (Thanks to Stefan Allan for this information.)
|- A = A
Theorem	eqidd 1519	Class identity law with antecedent.
|- (ph -> A = A)
Theorem	eqcom 1520	Commutative law for class equality. Theorem 6.5 of [Quine] p. 41.
|- (A = B <-> B = A)
Theorem	eqcoms 1521	Inference applying commutative law for class equality to an antecedent.
|- (A = B -> ph)   =>   |- (B = A -> ph)
Theorem	eqcomi 1522	Inference from commutative law for class equality.
|- A = B   =>   |- B = A
Theorem	eqcomd 1523	Deduction from commutative law for class equality.
|- (ph -> A = B)   =>   |- (ph -> B = A)
Theorem	eqeq1 1524	Equality implies equivalence of equalities.
|- (A = B -> (A = C <-> B = C))
Theorem	eqeq1i 1525	Inference from equality to equivalence of equalities.
|- A = B   =>   |- (A = C <-> B = C)
Theorem	eqeq1d 1526	Deduction from equality to equivalence of equalities.
|- (ph -> A = B)   =>   |- (ph -> (A = C <-> B = C))
Theorem	eqeq2 1527	Equality implies equivalence of equalities.
|- (A = B -> (C = A <-> C = B))
Theorem	eqeq2i 1528	Inference from equality to equivalence of equalities.
|- A = B   =>   |- (C = A <-> C = B)
Theorem	eqeq2d 1529	Deduction from equality to equivalence of equalities.
|- (ph -> A = B)   =>   |- (ph -> (C = A <-> C = B))
Theorem	eqeq12 1530	Equality relationship among 4 classes.
|- ((A = B /\ C = D) -> (A = C <-> B = D))
Theorem	eqeq12i 1531	A useful inference for substituting definitions into an equality.
|- A = B   &   |- C = D   =>   |- (A = C <-> B = D)
Theorem	eqeq12d 1532	A useful inference for substituting definitions into an equality.
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> (A = C <-> B = D))
Theorem	eqeqan12d 1533	A useful inference for substituting definitions into an equality.
|- (ph -> A = B)   &   |- (ps -> C = D)   =>   |- ((ph /\ ps) -> (A = C <-> B = D))
Theorem	eqeqan12rd 1534	A useful inference for substituting definitions into an equality.
|- (ph -> A = B)   &   |- (ps -> C = D)   =>   |- ((ps /\ ph) -> (A = C <-> B = D))
Theorem	eqtr 1535	Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13.
|- ((A = B /\ B = C) -> A = C)
Theorem	eqtr2 1536	A transitive law for class equality.
|- ((A = B /\ A = C) -> B = C)
Theorem	eqtr3 1537	A transitive law for class equality.
|- ((A = C /\ B = C) -> A = B)
Theorem	eqtri 1538	An equality transitivity inference.
|- A = B   &   |- B = C   =>   |- A = C
Theorem	eqtr2i 1539	An equality transitivity inference.
|- A = B   &   |- B = C   =>   |- C = A
Theorem	eqtr3i 1540	An equality transitivity inference.
|- A = B   &   |- A = C   =>   |- B = C
Theorem	eqtr4i 1541	An equality transitivity inference.
|- A = B   &   |- C = B   =>   |- A = C
Theorem	3eqtri 1542	An inference from three chained equalities.
|- A = B   &   |- B = C   &   |- C = D   =>   |- A = D
Theorem	3eqtrri 1543	An inference from three chained equalities.
|- A = B   &   |- B = C   &   |- C = D   =>   |- D = A
Theorem	3eqtr2i 1544	An inference from three chained equalities.
|- A = B   &   |- C = B   &   |- C = D   =>   |- A = D
Theorem	3eqtr2ri 1545	An inference from three chained equalities.
|- A = B   &   |- C = B   &   |- C = D   =>   |- D = A
Theorem	3eqtr3i 1546	An inference from three chained equalities.
|- A = B   &   |- A = C   &   |- B = D   =>   |- C = D
Theorem	3eqtr3ri 1547	An inference from three chained equalities.
|- A = B   &   |- A = C   &   |- B = D   =>   |- D = C
Theorem	3eqtr4i 1548	An inference from three chained equalities.
|- A = B   &   |- C = A   &   |- D = B   =>   |- C = D
Theorem	3eqtr4ri 1549	An inference from three chained equalities.
|- A = B   &   |- C = A   &   |- D = B   =>   |- D = C
Theorem	eqtrd 1550	An equality transitivity deduction.
|- (ph -> A = B)   &   |- (ph -> B = C)   =>   |- (ph -> A = C)
Theorem	eqtr2d 1551	An equality transitivity deduction.
|- (ph -> A = B)   &   |- (ph -> B = C)   =>   |- (ph -> C = A)
Theorem	eqtr3d 1552	An equality transitivity equality deduction.
|- (ph -> A = B)   &   |- (ph -> A = C)   =>   |- (ph -> B = C)
Theorem	eqtr4d 1553	An equality transitivity equality deduction.
|- (ph -> A = B)   &   |- (ph -> C = B)   =>   |- (ph -> A = C)
Theorem	3eqtrd 1554	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> B = C)   &   |- (ph -> C = D)   =>   |- (ph -> A = D)
Theorem	3eqtrrd 1555	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> B = C)   &   |- (ph -> C = D)   =>   |- (ph -> D = A)
Theorem	3eqtr2d 1556	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> C = B)   &   |- (ph -> C = D)   =>   |- (ph -> A = D)
Theorem	3eqtr2rd 1557	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> C = B)   &   |- (ph -> C = D)   =>   |- (ph -> D = A)
Theorem	3eqtr3d 1558	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> A = C)   &   |- (ph -> B = D)   =>   |- (ph -> C = D)
Theorem	3eqtr3rd 1559	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> A = C)   &   |- (ph -> B = D)   =>   |- (ph -> D = C)
Theorem	3eqtr4d 1560	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> C = A)   &   |- (ph -> D = B)   =>   |- (ph -> C = D)
Theorem	3eqtr4rd 1561	A deduction from three chained equalities.
|- (ph -> A = B)   &   |- (ph -> C = A)   &   |- (ph -> D = B)   =>   |- (ph -> D = C)
Theorem	syl5eq 1562	An equality transitivity deduction.
|- (ph -> A = B)   &   |- C = A   =>   |- (ph -> C = B)
Theorem	syl5req 1563	An equality transitivity deduction.
|- (ph -> A = B)   &   |- C = A   =>   |- (ph -> B = C)
Theorem	syl5eqr 1564	An equality transitivity deduction.
|- (ph -> A = B)   &   |- A = C   =>   |- (ph -> C = B)
Theorem	syl5reqr 1565	An equality transitivity deduction.
|- (ph -> A = B)   &   |- A = C   =>   |- (ph -> B = C)
Theorem	syl6eq 1566	An equality transitivity deduction.
|- (ph -> A = B)   &   |- B = C   =>   |- (ph -> A = C)
Theorem	syl6req 1567	An equality transitivity deduction.
|- (ph -> A = B)   &   |- B = C   =>   |- (ph -> C = A)
Theorem	syl6eqr 1568	An equality transitivity deduction.
|- (ph -> A = B)   &   |- C = B   =>   |- (ph -> A = C)
Theorem	syl6reqr 1569	An equality transitivity deduction.
|- (ph -> A = B)   &   |- C = B   =>   |- (ph -> C = A)
Theorem	sylan9eq 1570	An equality transitivity deduction.
|- (ph -> A = B)   &   |- (ps -> B = C)   =>   |- ((ph /\ ps) -> A = C)
Theorem	sylan9req 1571	An equality transitivity deduction.
|- (ph -> B = A)   &   |- (ps -> B = C)   =>   |- ((ph /\ ps) -> A = C)
Theorem	sylan9eqr 1572	An equality transitivity deduction.
|- (ph -> A = B)   &   |- (ps -> B = C)   =>   |- ((ps /\ ph) -> A = C)
Theorem	3eqtr3g 1573	A chained equality inference, useful for converting from definitions.
|- (ph -> A = B)   &   |- A = C   &   |- B = D   =>   |- (ph -> C = D)
Theorem	3eqtr4g 1574	A chained equality inference, useful for converting to definitions.
|- (ph -> A = B)   &   |- C = A   &   |- D = B   =>   |- (ph -> C = D)
Theorem	3eqtr4a 1575	A chained equality inference, useful for converting to definitions.
|- A = B   &   |- (ph -> C = A)   &   |- (ph -> D = B)   =>   |- (ph -> C = D)
Theorem	eq2tri 1576	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 1577	Equality implies equivalence of membership.
|- (A = B -> (A e. C <-> B e. C))
Theorem	eleq2 1578	Equality implies equivalence of membership.
|- (A = B -> (C e. A <-> C e. B))
Theorem	eleq12 1579	Equality implies equivalence of membership.
|- ((A = B /\ C = D) -> (A e. C <-> B e. D))
Theorem	eleq1i 1580	Inference from equality to equivalence of membership.
|- A = B   =>   |- (A e. C <-> B e. C)
Theorem	eleq2i 1581	Inference from equality to equivalence of membership.
|- A = B   =>   |- (C e. A <-> C e. B)
Theorem	eleq12i 1582	Inference from equality to equivalence of membership.
|- A = B   &   |- C = D   =>   |- (A e. C <-> B e. D)
Theorem	eleq1d 1583	Deduction from equality to equivalence of membership.
|- (ph -> A = B)   =>   |- (ph -> (A e. C <-> B e. C))
Theorem	eleq2d 1584	Deduction from equality to equivalence of membership.
|- (ph -> A = B)   =>   |- (ph -> (C e. A <-> C e. B))
Theorem	eleq12d 1585	Deduction from equality to equivalence of membership.
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> (A e. C <-> B e. D))
Theorem	eleq1a 1586	A transitive-type law relating membership and equality.
|- (A e. B -> (C = A -> C e. B))
Theorem	eqeltri 1587	Substitution of equal classes into membership relation.
|- A = B   &   |- B e. C   =>   |- A e. C
Theorem	eqeltrri 1588	Substitution of equal classes into membership relation.
|- A = B   &   |- A e. C   =>   |- B e. C
Theorem	eleqtri 1589	Substitution of equal classes into membership relation.
|- A e. B   &   |- B = C   =>   |- A e. C
Theorem	eleqtrri 1590	Substitution of equal classes into membership relation.
|- A e. B   &   |- C = B   =>   |- A e. C
Theorem	eqeltrd 1591	Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)
|- (ph -> A = B)   &   |- (ph -> B e. C)   =>   |- (ph -> A e. C)
Theorem	eqeltrrd 1592	Deduction that substitutes equal classes into membership.
|- (ph -> A = B)   &   |- (ph -> A e. C)   =>   |- (ph -> B e. C)
Theorem	eleqtrd 1593	Deduction that substitutes equal classes into membership.
|- (ph -> A e. B)   &   |- (ph -> B = C)   =>   |- (ph -> A e. C)
Theorem	eleqtrrd 1594	Deduction that substitutes equal classes into membership.
|- (ph -> A e. B)   &   |- (ph -> C = B)   =>   |- (ph -> A e. C)
Theorem	syl5eqel 1595	A membership and equality inference.
|- (ph -> A e. B)   &   |- C = A   =>   |- (ph -> C e. B)
Theorem	syl5eqelr 1596	A membership and equality inference.
|- (ph -> A e. B)   &   |- A = C   =>   |- (ph -> C e. B)
Theorem	syl5eleq 1597	A membership and equality inference.
|- (ph -> A = B)   &   |- C e. A   =>   |- (ph -> C e. B)
Theorem	syl5eleqr 1598	A membership and equality inference.
|- (ph -> B = A)   &   |- C e. A   =>   |- (ph -> C e. B)
Theorem	syl6eqel 1599	A membership and equality inference.
|- (ph -> A = B)   &   |- B e. C   =>   |- (ph -> A e. C)
Theorem	syl6eqelr 1600	A membership and equality inference.
|- (ph -> B = A)   &   |- B e. C   =>   |- (ph -> A e. C)