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Axiom ax-ext 2601
Description: 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 (∀𝑤(𝑤𝑥𝑤𝑦) → (𝑥𝑧𝑦𝑧)), and equality 𝑥 = 𝑦 is defined as 𝑤(𝑤𝑥𝑤𝑦). 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 predicate calculus axioms, we would rewrite all axioms involving equality with equality expanded according to the above definition. Some of those axioms may be provable from ax-ext and would become redundant, but this hasn't been studied carefully.

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 𝑥 in ax-ext 2601 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 4731, which involves a wff metavariable). In practice, though, the theorems and proofs are essentially the same. The $d restrictions make each of the infinite axioms generated by the ax-ext 2601 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 21-May-1993.)

Ref Expression
ax-ext (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Axiom ax-ext
StepHypRef Expression
1 vz . . . . 5 setvar 𝑧
2 vx . . . . 5 setvar 𝑥
31, 2wel 1988 . . . 4 wff 𝑧𝑥
4 vy . . . . 5 setvar 𝑦
51, 4wel 1988 . . . 4 wff 𝑧𝑦
63, 5wb 196 . . 3 wff (𝑧𝑥𝑧𝑦)
76, 1wal 1478 . 2 wff 𝑧(𝑧𝑥𝑧𝑦)
82, 4weq 1871 . 2 wff 𝑥 = 𝑦
97, 8wi 4 1 wff (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
Colors of variables: wff setvar class
This axiom is referenced by:  axext2  2602  axext3  2603  axext3ALT  2604  bm1.1  2606  ax6vsep  4745  nfnid  4857  bj-axext3  32409  bj-ax9  32534  bj-ax9-2  32535  axc11next  38086
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