Theorem zfreg 8113

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 8218). (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 8112	. 2 |- ( E. x x e. A -> E. x e. A A. y e. x -. y e. A )
3		n0 3771	. 2 |- ( A =/= (/) <-> E. x x e. A )
4		disj 3833	. . 3 |- ( ( x i^i A ) = (/) <-> A. y e. x -. y e. A )
5	4	rexbii 2927	. 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 269	1 |- ( A =/= (/) -> E. x e. A ( x i^i A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1437   E.wex 1659   e. wcel 1868   =/= wne 2618   A.wral 2775   E.wrex 2776   _Vcvv 3081   i^i cin 3435   (/)c0 3761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-reg 8110

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-v 3083  df-dif 3439  df-in 3443  df-nul 3762

This theorem is referenced by:  en3lp  8124  inf3lem3  8138  setindtr  35799