Theorem zfreg 4576

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 4627).

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

Step	Hyp	Ref	Expression
1		zfreg.1	. . 3 |- A e. V
2	1	zfregcl 4575	. 2 |- (E.x x e. A -> E.x e. A A.y e. x -. y e. A)
3		ne0 2284	. 2 |- (A =/= (/) <-> E.x x e. A)
4		disj 2307	. . 3 |- ((x i^i A) = (/) <-> A.y e. x -. y e. A)
5	4	rexbii 1665	. 2 |- (E.x e. A (x i^i A) = (/) <-> E.x e. A A.y e. x -. y e. A)
6	2, 3, 5	3imtr4 219	1 |- (A =/= (/) -> E.x e. A (x i^i A) = (/))

Syntax hints:  -. wn 2   -> wi 3   = wceq 954   e. wcel 956  E.wex 978   =/= wne 1582  A.wral 1642  E.wrex 1643  Vcvv 1807   i^i cin 2042  (/)c0 2276

This theorem is referenced by:  inf3lem3 4595

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-reg 4573

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-in 2047  df-nul 2277

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