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Mirrors > Home > MPE Home > Th. List > ax-sep | Structured version Visualization version GIF version |
Description: Axiom scheme of
separation. This is an axiom scheme of Zermelo and
Zermelo-Fraenkel set theories.
It was derived as axsep 5167 above and is therefore redundant in ZF set theory, which contains ax-rep 5155 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 3719. 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 5170, 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 5228 shows (showing the necessity of that condition in zfauscl 5170). Scheme Sep of [BellMachover] p. 463. (Contributed by NM, 11-Sep-2006.) |
Ref | Expression |
---|---|
ax-sep | ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vx | . . . . 5 setvar 𝑥 | |
2 | vy | . . . . 5 setvar 𝑦 | |
3 | 1, 2 | wel 2112 | . . . 4 wff 𝑥 ∈ 𝑦 |
4 | vz | . . . . . 6 setvar 𝑧 | |
5 | 1, 4 | wel 2112 | . . . . 5 wff 𝑥 ∈ 𝑧 |
6 | wph | . . . . 5 wff 𝜑 | |
7 | 5, 6 | wa 399 | . . . 4 wff (𝑥 ∈ 𝑧 ∧ 𝜑) |
8 | 3, 7 | wb 209 | . . 3 wff (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
9 | 8, 1 | wal 1536 | . 2 wff ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
10 | 9, 2 | wex 1781 | 1 wff ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
Colors of variables: wff setvar class |
This axiom is referenced by: axsepg 5169 zfauscl 5170 bm1.3ii 5171 ax6vsep 5172 axnul 5174 nalset 5182 bj-zfauscl 34386 bj-bm1.3ii 34500 |
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