Theorem zfreg 4568

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 4619).

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 4567	. 2 |- (E.x x e. A -> E.x e. A A.y e. x -. y e. A)
3		ne0 2278	. 2 |- (A =/= (/) <-> E.x x e. A)
4		disj 2301	. . 3 |- ((x i^i A) = (/) <-> A.y e. x -. y e. A)
5	4	rexbii 1660	. 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 953   e. wcel 955  E.wex 977   =/= wne 1577  A.wral 1637  E.wrex 1638  Vcvv 1802   i^i cin 2036  (/)c0 2270

This theorem is referenced by:  inf3lem3 4587

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-reg 4565

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-in 2041  df-nul 2271

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