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Mirrors > Home > MPE Home > Th. List > ax-reg | Structured version Visualization version GIF version |
Description: Axiom of Regularity. An axiom of Zermelo-Fraenkel set theory. Also called the Axiom of Foundation. A rather non-intuitive axiom that denies more than it asserts, it states (in the form of zfreg 9211) that every nonempty set contains a set disjoint from itself. One consequence is that it denies the existence of a set containing itself (elirrv 9212). A stronger version that works for proper classes is proved as zfregs 9348. (Contributed by NM, 14-Aug-1993.) |
Ref | Expression |
---|---|
ax-reg | ⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vy | . . . 4 setvar 𝑦 | |
2 | vx | . . . 4 setvar 𝑥 | |
3 | 1, 2 | wel 2111 | . . 3 wff 𝑦 ∈ 𝑥 |
4 | 3, 1 | wex 1787 | . 2 wff ∃𝑦 𝑦 ∈ 𝑥 |
5 | vz | . . . . . . 7 setvar 𝑧 | |
6 | 5, 1 | wel 2111 | . . . . . 6 wff 𝑧 ∈ 𝑦 |
7 | 5, 2 | wel 2111 | . . . . . . 7 wff 𝑧 ∈ 𝑥 |
8 | 7 | wn 3 | . . . . . 6 wff ¬ 𝑧 ∈ 𝑥 |
9 | 6, 8 | wi 4 | . . . . 5 wff (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥) |
10 | 9, 5 | wal 1541 | . . . 4 wff ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥) |
11 | 3, 10 | wa 399 | . . 3 wff (𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)) |
12 | 11, 1 | wex 1787 | . 2 wff ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)) |
13 | 4, 12 | wi 4 | 1 wff (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) |
Colors of variables: wff setvar class |
This axiom is referenced by: axreg2 9209 |
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