Theorem zfreg 7519

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 7624). (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 7518	. 2 |- ( E. x x e. A -> E. x e. A A. y e. x -. y e. A )
3		n0 3597	. 2 |- ( A =/= (/) <-> E. x x e. A )
4		disj 3628	. . 3 |- ( ( x i^i A ) = (/) <-> A. y e. x -. y e. A )
5	4	rexbii 2691	. 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 258	1 |- ( A =/= (/) -> E. x e. A ( x i^i A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   E.wex 1547   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   E.wrex 2667   _Vcvv 2916   i^i cin 3279   (/)c0 3588

This theorem is referenced by:  inf3lem3  7541  en3lp  7628  setindtr  26985

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-reg 7516

This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-v 2918  df-dif 3283  df-in 3287  df-nul 3589