mmtheorems15 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 1401-1500 - Page 15 of 105

Type	Label	Description
Statement
Theorem	moor 1401	"At most one" is still the case when a disjunct is removed.
|- (E*x(ph \/ ps) -> E*xph)
Theorem	mooran1 1402	"At most one" imports disjunction to conjunction.
|- ((E*xph \/ E*xps) -> E*x(ph /\ ps))
Theorem	mooran2 1403	"At most one" exports disjunction to conjunction.
|- (E*x(ph \/ ps) -> (E*xph /\ E*xps))
Theorem	moanim 1404	Introduction of a conjunct into "at most one" quantifier.
|- (ph -> A.xph)   =>   |- (E*x(ph /\ ps) <-> (ph -> E*xps))
Theorem	euan 1405	Introduction of a conjunct into uniqueness quantifier.
|- (ph -> A.xph)   =>   |- (E!x(ph /\ ps) <-> (ph /\ E!xps))
Theorem	moanimv 1406	Introduction of a conjunct into "at most one" quantifier.
|- (E*x(ph /\ ps) <-> (ph -> E*xps))
Theorem	moaneu 1407	Nested "at most one" and uniqueness quantifiers.
|- E*x(ph /\ E!xph)
Theorem	moanmo 1408	Nested "at most one" quantifiers.
|- E*x(ph /\ E*xph)
Theorem	euanv 1409	Introduction of a conjunct into uniqueness quantifier.
|- (E!x(ph /\ ps) <-> (ph /\ E!xps))
Theorem	mopick 1410	"At most one" picks a variable value, eliminating an existential quantifier.
|- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))
Theorem	eupick 1411	Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing x such that ph is true, and there is also an x (actually the same one) such that ph and ps are both true, then ph implies ps regardless of x. This theorem, which apparently does not appear explicitly in the literature, can be quite useful because it lets us eliminate existential quantifiers in a hypothesis.
|- ((E!xph /\ E.x(ph /\ ps)) -> (ph -> ps))
Theorem	eupickb 1412	Existential uniqueness "pick" showing wff equivalence.
|- ((E!xph /\ E!xps /\ E.x(ph /\ ps)) -> (ph <-> ps))
Theorem	mopick2 1413	"At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1070.
|- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> E.x(ph /\ ps /\ ch))
Theorem	euor2 1414	Introduce or eliminate a disjunct in a uniqueness quantifier.
|- (-. E.xph -> (E!x(ph \/ ps) <-> E!xps))
Theorem	moexex 1415	"At most one" double quantification.
|- (ph -> A.yph)   =>   |- ((E*xph /\ A.xE*yps) -> E*yE.x(ph /\ ps))
Theorem	moexexv 1416	"At most one" double quantification.
|- ((E*xph /\ A.xE*yps) -> E*yE.x(ph /\ ps))
Theorem	2moex 1417	Double quantification with "at most one."
|- (E*xE.yph -> A.yE*xph)
Theorem	2euex 1418	Double quantification with existential uniqueness.
|- (E!xE.yph -> E.yE!xph)
Theorem	2eumo 1419	Double quantification with existential uniqueness and "at most one."
|- (E!xE*yph -> E*xE!yph)
Theorem	2eu2ex 1420	Double existential uniqueness.
|- (E!xE!yph -> E.xE.yph)
Theorem	2moswap 1421	A condition allowing swap of "at most one" and existential quantifiers.
|- (A.xE*yph -> (E*xE.yph -> E*yE.xph))
Theorem	2euswap 1422	A condition allowing swap of uniqueness and existential quantifiers.
|- (A.xE*yph -> (E!xE.yph -> E!yE.xph))
Theorem	2exeu 1423	Double existential uniqueness implies double uniqueness quantification.
|- ((E!xE.yph /\ E!yE.xph) -> E!xE!yph)
Theorem	2mo 1424	Two equivalent expressions for double "at most one."
|- (E.zE.wA.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)))
Theorem	2mos 1425	Double "exists at most one" with implicit substitution.
|- ((x = z /\ y = w) -> (ph <-> ps))   =>   |- (E.zE.wA.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.yA.zA.w((ph /\ ps) -> (x = z /\ y = w)))
Theorem	2eu1 1426	Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one.
|- (A.xE*yph -> (E!xE!yph <-> (E!xE.yph /\ E!yE.xph)))
Theorem	2eu2 1427	Double existential uniqueness.
|- (E!yE.xph -> (E!xE!yph <-> E!xE.yph))
Theorem	2eu3 1428	Double existential uniqueness.
|- (A.xA.y(E*xph \/ E*yph) -> ((E!xE!yph /\ E!yE!xph) <-> (E!xE.yph /\ E!yE.xph)))
Theorem	2eu4 1429	This theorem provides us with a definition of double existential uniqueness ("exactly one x and exactly one y"). Naively one might think (incorrectly) that it could be defined by E!xE!yph. See 2eu1 1426 for a condition under which the naive definition holds and 2exeu 1423 for a one-way implication. See 2eu5 1430 and 2eu8 1433 for alternate definitions.
|- ((E!xE.yph /\ E!yE.xph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))
Theorem	2eu5 1430	An alternate definition of double existential uniqueness (see 2eu4 1429). A mistake sometimes made in the literature is to use E!xE!y to mean "exactly one x and exactly one y." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining A.xE*yph as an additional condition. The correct definition apparently has never been published. (E* means "exists at most one.")
|- ((E!xE!yph /\ A.xE*yph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))
Theorem	2eu6 1431	Two equivalent expressions for double existential uniqueness.
|- ((E!xE.yph /\ E!yE.xph) <-> E.zE.wA.xA.y(ph <-> (x = z /\ y = w)))
Theorem	2eu7 1432	Two equivalent expressions for double existential uniqueness.
|- ((E!xE.yph /\ E!yE.xph) <-> E!xE!y(E.xph /\ E.yph))
Theorem	2eu8 1433	Two equivalent expressions for double existential uniqueness. Curiously, we can put E! on either of the internal conjuncts but not both. We can also commute E!xE!y using 2eu7 1432.
|- (E!xE!y(E.xph /\ E.yph) <-> E!xE!y(E!xph /\ E.yph))
Theorem	exists1 1434	Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru 2740.
|- (E!x x = x <-> A.x x = y)
Theorem	exists2 1435	A condition implying that at least two things exist.
|- ((E.xph /\ E.x -. ph) -> -. E!x x = x)
ZF Set Theory - start with the Axiom of Extensionality
Introduce the Axiom of Extensionality
Axiom	ax-ext 1436	Axiom of Extensionality. An axiom of Zermelo-Fraenkel set theory. It states that two sets are identical if they contain the same elements. Axiom Ext of [BellMachover] p. 461. Set theory can also be formulated with a single primitive predicate e. on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes (A.w(w e. x <-> w e. y) -> (x e. z -> y e. z)), and equality x = y is defined as A.w(w e. x <-> w e. y). All of the usual axioms of equality then become theorems of set theory. See, for example, Axiom 1 of [TakeutiZaring] p. 8. To use the above "equality-free" version of Extensionality with Metamath's logical axioms, we would rewrite ax-8 1101 through ax-16 1194 with equality expanded according to the above definition. Some of those axioms could be proved from set theory and would be redundant. Not all of them are redundant, since our axioms of predicate calculus make essential use of equality for the proper substitution that is a primitive notion in traditional predicate calculus. A study of such an axiomatization would be an interesting project for someone exploring the foundations of logic. General remarks: Our set theory axioms are presented using defined connectives (<->, E., etc.) for convenience. However, it is implicitly understood that the actual axioms use only the primitive connectives ->, -., A., =, and e.. It is straightforward to establish the equivalence between the actual axioms and the ones we display, and we will not do so. It is important to understand that strictly speaking, all of our set theory axioms are really schemes that represent an infinite number of actual axioms. This is inherent in the design of Metamath ("metavariable math"), which manipulates only metavariables. For example, the metavariable x in ax-ext 1436 can represent any actual variable v1, v2, v3,... . Distinct variable restrictions ($d) prevent us from substituting say v1 for both x and z. This is in contrast to typical textbook presentations that present actual axioms (except for Replacement ax-rep 2661, which involves a wff metavariable). In practice, though, the theorems and proofs are essentially the same. The $d restrictions make each of the the infinite axioms generated by the ax-ext 1436 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version.
|- (A.z(z e. x <-> z e. y) -> x = y)
Theorem	axext 1437	The Axiom of Extensionality (ax-ext 1436) 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	zfext2 1438	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	bm1.1 1439	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 1440	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 2252). Note that a set variable can be expressed as a class builder per theorem cvjust 1448, justifying the assignment of set variables to class variables via the use of cv 1098.
class {x | ph}
Definition	df-clab 1441	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 1105, which extends or "overloads" the wel 1106 definition connecting set variables, requires that both sides of e. be a class. In df-cleq 1446 and df-clel 1449, 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 1098 allows us to substitute a set variable x for a class variable: all sets are classes by cvjust 1448 (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 1544 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 1822 which is used, for example, to convert elirrv 4522 to elirr 4523.
|- (x e. {y | ph} <-> [x / y]ph)
Theorem	abid 1442	Simplification of class abstraction notation when the free and bound variables are identical.
|- (x e. {x | ph} <-> ph)
Theorem	hbab1 1443	Bound-variable hypothesis builder for a class abstraction.
|- (y e. {x | ph} -> A.x y e. {x | ph})
Theorem	hbab 1444	Bound-variable hypothesis builder for a class abstraction.
|- (ph -> A.xph)   =>   |- (z e. {y | ph} -> A.x z e. {y | ph})
Theorem	hbabd 1445	Deduction form of bound-variable hypothesis builder hbab 1444.
|- (ph -> A.xA.yph)   &   |- (ph -> (ps -> A.xps))   =>   |- (ph -> (z e. {y | ps} -> A.x z e. {y | ps}))
Definition	df-cleq 1446	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. 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 1441, df-clel 1449, and abeq2 1544.
|- (A.x(x e. y <-> x e. z) -> y = z)   =>   |- (A = B <-> A.x(x e. A <-> x e. B))
Theorem	dfcleq 1447	The same as df-cleq 1446 with the hypothesis removed using the Axiom of Extensionality ax-ext 1436.
|- (A = B <-> A.x(x e. A <-> x e. B))
Theorem	cvjust 1448	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 1098, which allows us to substitute a set variable for a class variable. See also cab 1440 and df-clab 1441. Note that this is not a rigorous justification, because cv 1098 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 1449	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 1446 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 1446 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 1310), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 1441.
|- (A e. B <-> E.x(x = A /\ x e. B))
Theorem	eqrdv 1450	Deduce equality of classes from equivalence of membership.
|- (ph -> (x e. A <-> x e. B))   =>   |- (ph -> A = B)
Theorem	eqriv 1451	Infer equality of classes from equivalence of membership.
|- (x e. A <-> x e. B)   =>   |- A =