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Mirrors > Home > MPE Home > Th. List > ax-sep | Structured version Visualization version GIF version |
Description: The Axiom of Separation of ZF set theory. See axsep 4708 for more information. It was derived as axsep 4708 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. (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 1978 | . . . 4 wff 𝑥 ∈ 𝑦 |
4 | vz | . . . . . 6 setvar 𝑧 | |
5 | 1, 4 | wel 1978 | . . . . 5 wff 𝑥 ∈ 𝑧 |
6 | wph | . . . . 5 wff 𝜑 | |
7 | 5, 6 | wa 383 | . . . 4 wff (𝑥 ∈ 𝑧 ∧ 𝜑) |
8 | 3, 7 | wb 195 | . . 3 wff (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
9 | 8, 1 | wal 1473 | . 2 wff ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
10 | 9, 2 | wex 1695 | 1 wff ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
Colors of variables: wff setvar class |
This axiom is referenced by: axsep2 4710 zfauscl 4711 bm1.3ii 4712 ax6vsep 4713 axnul 4716 nalset 4723 bj-nalset 31982 bj-axsep2 32113 |
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