Theorem zfreg 9043

Description: The Axiom of Regularity using abbreviations. Axiom 6 of [TakeutiZaring] p. 21. This is called the "weak form". Axiom Reg of [BellMachover] p. 480. There is also a "strong form", not requiring that 𝐴 be a set, that can be proved with more difficulty (see zfregs 9158). (Contributed by NM, 26-Nov-1995.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
zfreg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem zfreg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 4260	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2	1	biimpi 219	. . 3 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)
3	2	anim2i 619	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (𝐴 ∈ 𝑉 ∧ ∃𝑥 𝑥 ∈ 𝐴))
4		zfregcl 9042	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴))
5	4	imp 410	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴)
6		disj 4355	. . . 4 ⊢ ((𝑥 ∩ 𝐴) = ∅ ↔ ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴)
7	6	rexbii 3210	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴)
8	7	biimpri 231	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
9	3, 5, 8	3syl 18	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   ∩ cin 3880  ∅c0 4243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770  ax-reg 9040

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-rex 3112  df-dif 3884  df-in 3888  df-nul 4244

This theorem is referenced by:  zfregfr  9052  en3lp  9061  inf3lem3  9077  bj-restreg  34514  setindtr  39965