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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 5202 above and is therefore redundant in ZF set theory, which contains ax-rep 5190 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 3771. 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 5205, 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 5262 shows (showing the necessity of that condition in zfauscl 5205). 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 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 398 | . . . 4 wff (𝑥 ∈ 𝑧 ∧ 𝜑) |
8 | 3, 7 | wb 208 | . . 3 wff (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
9 | 8, 1 | wal 1535 | . 2 wff ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
10 | 9, 2 | wex 1780 | 1 wff ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
Colors of variables: wff setvar class |
This axiom is referenced by: axsepg 5204 zfauscl 5205 bm1.3ii 5206 ax6vsep 5207 axnul 5209 nalset 5217 bj-zfauscl 34246 bj-bm1.3ii 34360 |
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