Theorem zfreg 3447

Description: The Axiom of Regularity using abbreviations. Axiom 6 of [TakeutiZaring] p. 21. This is called the 'weak form'; there is also a 'strong form', not requiring that A be a set, that can be proved with more difficulty (see zfregs 3491).

Hypothesis

Ref	Expression
zfreg.1	⊢ A ∈ V

Assertion

Ref	Expression
zfreg	⊢ (¬ A = ∅ → ∃x ∈ A (x ∩ A) = ∅)

Distinct variable group(s):   x,A

Proof of Theorem zfreg

Step	Hyp	Ref	Expression
1		zfreg.1	. . 3 ⊢ A ∈ V
2	1	zfregcl 3446	. 2 ⊢ (∃x x ∈ A → ∃x ∈ A ∀y ∈ x ¬ y ∈ A)
3		n0 1714	. 2 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
4		disj 1733	. . 3 ⊢ ((x ∩ A) = ∅ ↔ ∀y ∈ x ¬ y ∈ A)
5	4	birex 1224	. 2 ⊢ (∃x ∈ A (x ∩ A) = ∅ ↔ ∃x ∈ A ∀y ∈ x ¬ y ∈ A)
6	2, 3, 5	3imtr4 192	1 ⊢ (¬ A = ∅ → ∃x ∈ A (x ∩ A) = ∅)

Syntax hints:  ¬ wn 1   → wi 2  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   ∩ cin 1486  ∅c0 1707

This theorem is referenced by:  inf3lem3 3466

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074  ax-reg 1078

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-in 1491  df-nul 1708

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