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Axiom ax-reg 8441
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 8444) that every nonempty set contains a set disjoint from itself. One consequence is that it denies the existence of a set containing itself (elirrv 8448). A stronger version that works for proper classes is proved as zfregs 8552. (Contributed by NM, 14-Aug-1993.)
Assertion
Ref Expression
ax-reg (∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Axiom ax-reg
StepHypRef Expression
1 vy . . . 4 setvar 𝑦
2 vx . . . 4 setvar 𝑥
31, 2wel 1988 . . 3 wff 𝑦𝑥
43, 1wex 1701 . 2 wff 𝑦 𝑦𝑥
5 vz . . . . . . 7 setvar 𝑧
65, 1wel 1988 . . . . . 6 wff 𝑧𝑦
75, 2wel 1988 . . . . . . 7 wff 𝑧𝑥
87wn 3 . . . . . 6 wff ¬ 𝑧𝑥
96, 8wi 4 . . . . 5 wff (𝑧𝑦 → ¬ 𝑧𝑥)
109, 5wal 1478 . . . 4 wff 𝑧(𝑧𝑦 → ¬ 𝑧𝑥)
113, 10wa 384 . . 3 wff (𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥))
1211, 1wex 1701 . 2 wff 𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥))
134, 12wi 4 1 wff (∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))
Colors of variables: wff setvar class
This axiom is referenced by:  axreg2  8442
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