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**Statement List for Metamath Proof Explorer -** 1401-1500 - Page 15 of 123
Type	Label	Description
Statement

Theorem	dveel2ALT 1401	Version of dveel2 1396 using ax-16 1247 instead of ax-17 1007.


Theorem	ax11eq 1402	Basis step for constructing a substitution instance of ax-11o 1255 without using ax-11o 1255. Atomic formula for equality predicate.


Theorem	ax11el 1403	Basis step for constructing a substitution instance of ax-11o 1255 without using ax-11o 1255. Atomic formula for membership predicate.


Theorem	ax11f 1404	Basis step for constructing a substitution instance of ax-11o 1255 without using ax-11o 1255. We can start with any formula in which is not free.


Theorem	ax11indn 1405	Induction step for constructing a substitution instance of ax-11o 1255 without using ax-11o 1255. Negation case.


Theorem	ax11indi 1406	Induction step for constructing a substitution instance of ax-11o 1255 without using ax-11o 1255. Implication case.


Theorem	ax11indalem 1407	Lemma for ax11inda2 1409 and ax11inda 1410.

Theorem	ax11inda2ALT 1408	A proof of ax11inda2 1409 that is slightly more direct.


Theorem	ax11inda2 1409	Induction step for constructing a substitution instance of ax-11o 1255 without using ax-11o 1255. Quantification case. When and are distinct, this theorem avoids the dummy variables needed by the more general ax11inda 1410.


Theorem	ax11inda 1410	Induction step for constructing a substitution instance of ax-11o 1255 without using ax-11o 1255. Quantification case. (When and are distinct, ax11inda2 1409 may be used instead to avoid the dummy variable in the proof.)


Theorem	a12stdy1 1411	Part of a study related to ax-12 1004. The consequent introduces a new variable . There are no distinct variable restrictions.


Theorem	a12stdy2 1412	Part of a study related to ax-12 1004. The consequent is quantified with a different variable. There are no distinct variable restrictions.
$/\$

Theorem	a12stdy3 1413	Part of a study related to ax-12 1004. The consequent introduces two new variables. There are no distinct variable restrictions.
$/\$

Theorem	a12stdy4 1414	Part of a study related to ax-12 1004. The second antecedent of ax-12 1004 is replaced. There are no distinct variable restrictions.


Theorem	a12lem1 1415	Proof of first hypothesis of a12study 1417.


Theorem	a12lem2 1416	Proof of second hypothesis of a12study 1417.


Theorem	a12study 1417	Rederivation of axiom ax-12 1004 from two shorter formulas, without using ax-12 1004. See a12lem1 1415 and a12lem2 1416 for the proofs of the hypotheses (using ax-12 1004). This is the only known breakdown of ax-12 1004 into shorter formulas. See a12studyALT 1418 for an alternate proof. Note that the proof depends on ax-11o 1255, whose proof ax11o 1254 depends on ax-12 1004, meaning that we would have to replace ax-11 1003 with ax-11o 1255 in an axiomatization that uses the hypotheses in place of ax-12 1004. Whether this can be avoided is an open problem.


Theorem	a12studyALT 1418	Alternate proof of a12study 1417, also without using ax-12 1004.


Existential uniqueness

Syntax	weu 1419	Extend wff definition to include existential uniqueness ("there exists a unique such that ").


Syntax	wmo 1420	Extend wff definition to include uniqueness ("there exists at most one such that ").


Definition	df-eu 1421	Define existential uniqueness, i.e. "there exists exactly one such that ." Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 1431, eu2 1435, eu3 1436, and eu5 1448 (which in some cases we show with a hypothesis in place of a distinct variable condition on and ). Double uniqueness is tricky: does not mean "exactly one and one " (see 2eu4 1492).


Definition	df-mo 1422	Define "there exists at most one such that ." Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 1440. For other possible definitions see mo2 1439 and mo4 1442.


Theorem	euf 1423	A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition.


Theorem	eubid 1424	Formula-building rule for uniqueness quantifier (deduction rule).


Theorem	eubidv 1425	Formula-building rule for uniqueness quantifier (deduction rule).


Theorem	eubii 1426	Introduce uniqueness quantifier to both sides of an equivalence.


Theorem	hbeu1 1427	Bound-variable hypothesis builder for uniqueness.


Theorem	hbeu 1428	Bound-variable hypothesis builder for "at most one." Note that and needn't be distinct (this makes the proof more difficult).


Theorem	sb8eu 1429	Variable substitution in uniqueness quantifier. (This theorem can also be proved without requiring that and be distinct, but the proof would be longer.)


Theorem	cbveu 1430	Rule used to change bound variables, using implicit substitition.


Theorem	eu1 1431	An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110.
$/\$

Theorem	mo 1432	Equivalent definitions of "there exists at most one."
$/\$

Theorem	euex 1433	Existential uniqueness implies existence.


Theorem	eumo0 1434	Existential uniqueness implies "at most one."


Theorem	eu2 1435	An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26.
$/\$ $/\$

Theorem	eu3 1436	An alternate way to express existential uniqueness.
$/\$

Theorem	euor 1437	Introduce a disjunct into a uniqueness quantifier.
$/\$ $\/$

Theorem	euorv 1438	Introduce a disjunct into a uniqueness quantifier.
$/\$ $\/$

Theorem	mo2 1439	Alternate definition of "at most one."


Theorem	mo3 1440	Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that not occur in in place of our hypothesis.
$/\$

Theorem	mo4f 1441	"At most one" expressed using implicit substitution.
$/\$

Theorem	mo4 1442	"At most one" expressed using implicit substitution.
$/\$

Theorem	mobid 1443	Formula-building rule for "at most one" quantifier (deduction rule).


Theorem	mobii 1444	Formula-building rule for "at most one" quantifier (inference rule).


Theorem	hbmo1 1445	Bound-variable hypothesis builder for "at most one."


Theorem	hbmo 1446	Bound-variable hypothesis builder for "at most one."


Theorem	cbvmo 1447	Rule used to change bound variables, using implicit substitition.


Theorem	eu5 1448	Uniqueness in terms of "at most one."
$/\$

Theorem	eu4 1449	Uniqueness using implicit substitution.
$/\$ $/\$

Theorem	eumo 1450	Existential uniqueness implies "at most one."


Theorem	eumoi 1451	"At most one" inferred from existential uniqueness.


Theorem	exmoeu 1452	Existence in terms of "at most one" and uniqueness.


Theorem	exmoeu2 1453	Existence implies "at most one" is equivalent to uniqueness.


Theorem	moabs 1454	Absorption of existence condition by "at most one."


Theorem	exmo 1455	Something exists or at most one exists.
$\/$

Theorem	immo 1456	"At most one" is preserved through implication (notice wff reversal).


Theorem	immoi 1457	"At most one" is preserved through implication (notice wff reversal).


Theorem	moimv 1458	Move antecedent outside of "at most one."


Theorem	euimmo 1459	Uniqueness implies "at most one" through implication.


Theorem	euim 1460	Add existential uniqueness quantifiers to an implication. Note the reversed implication in the antecedent.
$/\$

Theorem	moan 1461	"At most one" is still the case when a conjunct is added.
$/\$

Theorem	moani 1462	"At most one" is still true when a conjunct is added.
$/\$

Theorem	moor 1463	"At most one" is still the case when a disjunct is removed.
$\/$

Theorem	mooran1 1464	"At most one" imports disjunction to conjunction.
$\/$ $/\$

Theorem	mooran2 1465	"At most one" exports disjunction to conjunction.
$\/$ $/\$

Theorem	moanim 1466	Introduction of a conjunct into "at most one" quantifier.
$/\$

Theorem	euan 1467	Introduction of a conjunct into uniqueness quantifier.
$/\$ $/\$

Theorem	moanimv 1468	Introduction of a conjunct into "at most one" quantifier.
$/\$

Theorem	moaneu 1469	Nested "at most one" and uniqueness quantifiers.
$/\$

Theorem	moanmo 1470	Nested "at most one" quantifiers.
$/\$

Theorem	euanv 1471	Introduction of a conjunct into uniqueness quantifier.
$/\$ $/\$

Theorem	mopick 1472	"At most one" picks a variable value, eliminating an existential quantifier.
$/\$ $/\$

Theorem	eupick 1473	Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing such that is true, and there is also an (actually the same one) such that and are both true, then implies regardless of . 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.
$/\$ $/\$

Theorem	eupicka 1474	Version of eupick 1473 with closed formulas.
$/\$ $/\$

Theorem	eupickb 1475	Existential uniqueness "pick" showing wff equivalence.
$/\$ $/\$ $/\$

Theorem	mopick2 1476	"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 1130.
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	euor2 1477	Introduce or eliminate a disjunct in a uniqueness quantifier.
$\/$

Theorem	moexex 1478	"At most one" double quantification.
$/\$ $/\$

Theorem	moexexv 1479	"At most one" double quantification.
$/\$ $/\$

Theorem	2moex 1480	Double quantification with "at most one."


Theorem	2euex 1481	Double quantification with existential uniqueness.


Theorem	2eumo 1482	Double quantification with existential uniqueness and "at most one."


Theorem	2eu2ex 1483	Double existential uniqueness.


Theorem	2moswap 1484	A condition allowing swap of "at most one" and existential quantifiers.


Theorem	2euswap 1485	A condition allowing swap of uniqueness and existential quantifiers.


Theorem	2exeu 1486	Double existential uniqueness implies double uniqueness quantification.
$/\$

Theorem	2mo 1487	Two equivalent expressions for double "at most one."
$/\$ $/\$ $/\$

Theorem	2mos 1488	Double "exists at most one", using implicit substitition.
$/\$ $/\$ $/\$ $/\$

Theorem	2eu1 1489	Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one.
$/\$

Theorem	2eu2 1490	Double existential uniqueness.


Theorem	2eu3 1491	Double existential uniqueness.
$\/$ $/\$ $/\$

Theorem	2eu4 1492	This theorem provides us with a definition of double existential uniqueness ("exactly one and exactly one "). Naively one might think (incorrectly) that it could be defined by . See 2eu1 1489 for a condition under which the naive definition holds and 2exeu 1486 for a one-way implication. See 2eu5 1493 and 2eu8 1496 for alternate definitions.
$/\$ $/\$ $/\$

Theorem	2eu5 1493	An alternate definition of double existential uniqueness (see 2eu4 1492). A mistake sometimes made in the literature is to use to mean "exactly one and exactly one ." (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 as an additional condition. The correct definition apparently has never been published. ( means "exists at most one.")
$/\$ $/\$ $/\$

Theorem	2eu6 1494	Two equivalent expressions for double existential uniqueness.
$/\$ $/\$

Theorem	2eu7 1495	Two equivalent expressions for double existential uniqueness.
$/\$ $/\$

Theorem	2eu8 1496	Two equivalent expressions for double existential uniqueness. Curiously, we can put on either of the internal conjuncts but not both. We can also commute using 2eu7 1495.
$/\$ $/\$

Theorem	euequ1 1497	Equality has existential uniqueness. Special case of eueq1 1963 proved using only predicate calculus. (Contributed by Stefan Allan, 4-Dec-2008.)


Theorem	exists1 1498	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 2831.


Theorem	exists2 1499	A condition implying that at least two things exist.
$/\$

ZF Set Theory - start with the Axiom of Extensionality

Introduce the Axiom of Extensionality

Axiom	ax-ext 1500	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 on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes , and equality is defined as . 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 1000 through ax-16 1247 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 (, , etc.) for convenience. However, it is implicitly understood that the actual axioms use only the primitive connectives , , , , and . 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 in ax-ext 1500 can represent any actual variable v1, v2, v3,... . Distinct variable restrictions ($d) prevent us from substituting say v1 for both and . This is in contrast to typical textbook presentations that present actual axioms (except for Replacement ax-rep 2767, 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 1500 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version.

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