Axiom ax-ext 2709

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. Its converse is a theorem of predicate logic, elequ2g 2122.

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(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)), and equality 𝑥 = 𝑦 is defined as ∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 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 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 2709 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 5209, 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 2709 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 21-May-1993.)

Assertion

Ref	Expression
ax-ext	⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Axiom ax-ext

Step	Hyp	Ref	Expression
1		vz	. . . . 5 setvar 𝑧
2		vx	. . . . 5 setvar 𝑥
3	1, 2	wel 2107	. . . 4 wff 𝑧 ∈ 𝑥
4		vy	. . . . 5 setvar 𝑦
5	1, 4	wel 2107	. . . 4 wff 𝑧 ∈ 𝑦
6	3, 5	wb 205	. . 3 wff (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)
7	6, 1	wal 1537	. 2 wff ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)
8	2, 4	weq 1966	. 2 wff 𝑥 = 𝑦
9	7, 8	wi 4	1 wff (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)

This axiom is referenced by:  axexte  2710  axextg  2711  axextmo  2713  ax6vsep  5227  nfnid  5298  axc11next  42024