Axiom ax-sep 5252

Description: Axiom scheme of separation. This is an axiom scheme of Zermelo and Zermelo-Fraenkel set theories.

It was derived as axsep 5251 above and is therefore redundant in ZF set theory, which contains ax-rep 5235 as an axiom (contrary to Zermelo set theory). We state it as a separate axiom here so that some of its uses can be identified more easily. Some textbooks present the axiom scheme of separation as a separate axiom scheme in order to show that much of set theory can be derived without the stronger axiom scheme of replacement (which is not part of Zermelo set theory).

The axiom scheme of separation is a weak form of Frege's axiom scheme of (unrestricted) comprehension, in that it conditions it with the condition 𝑥 ∈ 𝑧, so that it asserts the existence of a collection only if it is smaller than some other collection 𝑧 that already exists. This prevents Russell's paradox ru 3741. In some texts, this scheme is called "Aussonderung" (German for "separation") or "Subset Axiom".

The variable 𝑥 can occur in the formula 𝜑, which in textbooks is often written 𝜑(𝑥). To specify this in the Metamath language, we omit the distinct variable condition ($d) that 𝑥 not occur in 𝜑.

For a version using a class variable, see sepg 5254, which requires the axiom of extensionality as well as the axiom scheme of separation for its derivation.

If we omit the requirement that 𝑦 not occur in 𝜑, we can derive a contradiction, as notsep 5329 shows (showing the necessity of that condition in sepgi 5255, hence in sepg 5254 and ax-sep 5252).

Scheme Sep of [BellMachover] p. 463. (Contributed by NM, 11-Sep-2006.)

Assertion

Ref	Expression
ax-sep	⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧

Allowed substitution hint:   𝜑(𝑥)

Detailed syntax breakdown of Axiom ax-sep

Step	Hyp	Ref	Expression
1		vx	. . . . 5 setvar 𝑥
2		vy	. . . . 5 setvar 𝑦
3	1, 2	wel 2148	. . . 4 wff 𝑥 ∈ 𝑦
4		vz	. . . . . 6 setvar 𝑧
5	1, 4	wel 2148	. . . . 5 wff 𝑥 ∈ 𝑧
6		wph	. . . . 5 wff 𝜑
7	5, 6	wa 401	. . . 4 wff (𝑥 ∈ 𝑧 ∧ 𝜑)
8	3, 7	wb 209	. . 3 wff (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
9	8, 1	wal 1568	. 2 wff ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
10	9, 2	wex 1812	1 wff ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

This axiom is used by:  axsepg  5253  sepg  5254  zfausclOLD  5256  sepexlem  5257  ax6vsep  5260  axnul  5262  exnelv  5270  nalsetOLD  5272  axsepg2  35907  axsepg3  35908  axsepg3ALT  35909  bj-sepg  38013  bj-bm1.3ii  38169  ssclaxsep  46165