Axiom ax-sep 5220

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

It was derived as axsep 5219 above and is therefore redundant in ZF set theory, which contains ax-rep 5201 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 3723. 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 zfauscl 5222, 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 notzfaus 5294 shows (showing the necessity of that condition in zfauscl 5222).

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 2115	. . . 4 wff 𝑥 ∈ 𝑦
4		vz	. . . . . 6 setvar 𝑧
5	1, 4	wel 2115	. . . . 5 wff 𝑥 ∈ 𝑧
6		wph	. . . . 5 wff 𝜑
7	5, 6	wa 395	. . . 4 wff (𝑥 ∈ 𝑧 ∧ 𝜑)
8	3, 7	wb 206	. . 3 wff (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
9	8, 1	wal 1540	. 2 wff ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
10	9, 2	wex 1781	1 wff ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

This axiom is referenced by:  axsepg  5221  zfauscl  5222  sepexlem  5223  bm1.3iiOLD  5226  ax6vsep  5227  axnul  5229  exnelv  5237  nalsetOLD  5239  axsepg2  35213  axsepg2ALT  35214  bj-zfauscl  37219  bj-bm1.3ii  37359  ssclaxsep  45397