Theorem zfreg 7802

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 7944). (Contributed by NM, 26-Nov-1995.)

Hypothesis

Ref	Expression
zfreg.1	|- A e. _V

Assertion

Ref	Expression
zfreg	|- ( A =/= (/) -> E. x e. A ( x i^i A ) = (/) )

Distinct variable group:   x, A

Proof of Theorem zfreg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfreg.1	. . 3 |- A e. _V
2	1	zfregcl 7801	. 2 |- ( E. x x e. A -> E. x e. A A. y e. x -. y e. A )
3		n0 3641	. 2 |- ( A =/= (/) <-> E. x x e. A )
4		disj 3714	. . 3 |- ( ( x i^i A ) = (/) <-> A. y e. x -. y e. A )
5	4	rexbii 2735	. 2 |- ( E. x e. A ( x i^i A ) = (/) <-> E. x e. A A. y e. x -. y e. A )
6	2, 3, 5	3imtr4i 266	1 |- ( A =/= (/) -> E. x e. A ( x i^i A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1369   E.wex 1586   e. wcel 1756   =/= wne 2601   A.wral 2710   E.wrex 2711   _Vcvv 2967   i^i cin 3322   (/)c0 3632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-reg 7799

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-v 2969  df-dif 3326  df-in 3330  df-nul 3633

This theorem is referenced by:  en3lp  7814  inf3lem3  7828  setindtr  29326