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 e. V /\ A =/= (/) ) -> E. x e. A ( x i^i 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 =/= (/) <-> E. x x e. A )
2	1	biimpi 206	. . 3 |- ( A =/= (/) -> E. x x e. A )
3	2	anim2i 593	. 2 |- ( ( A e. V /\ A =/= (/) ) -> ( A e. V /\ E. x x e. A ) )
4		zfregcl 8499	. . 3 |- ( A e. V -> ( E. x x e. A -> E. x e. A A. y e. x -. y e. A ) )
5	4	imp 445	. 2 |- ( ( A e. V /\ E. x x e. A ) -> E. x e. A A. y e. x -. y e. A )
6		disj 4017	. . . 4 |- ( ( x i^i A ) = (/) <-> A. y e. x -. y e. A )
7	6	rexbii 3041	. . 3 |- ( E. x e. A ( x i^i A ) = (/) <-> E. x e. A A. y e. x -. y e. A )
8	7	biimpri 218	. 2 |- ( E. x e. A A. y e. x -. y e. A -> E. x e. A ( x i^i A ) = (/) )
9	3, 5, 8	3syl 18	1 |- ( ( A e. V /\ A =/= (/) ) -> E. x e. A ( x i^i A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   i^i 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