Theorem zfreg 8500

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 8608). (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 3931	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2	1	biimpi 206	. . 3 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)
3	2	anim2i 593	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (𝐴 ∈ 𝑉 ∧ ∃𝑥 𝑥 ∈ 𝐴))
4		zfregcl 8499	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴))
5	4	imp 445	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴)
6		disj 4017	. . . 4 ⊢ ((𝑥 ∩ 𝐴) = ∅ ↔ ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴)
7	6	rexbii 3041	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅ ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴)
8	7	biimpri 218	. 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 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913   ∩ cin 3573  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916

This theorem is referenced by:  zfregfr  8509  en3lp  8513  inf3lem3  8527  bj-restreg  33052  setindtr  37591