Theorem zfreg 4596

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 4647).

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 4595	. 2 |- (E.x x e. A -> E.x e. A A.y e. x -. y e. A)
3		ne0 2288	. 2 |- (A =/= (/) <-> E.x x e. A)
4		disj 2311	. . 3 |- ((x i^i A) = (/) <-> A.y e. x -. y e. A)
5	4	rexbii 1668	. 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 956   e. wcel 958  E.wex 980   =/= wne 1585  A.wral 1645  E.wrex 1646  Vcvv 1811   i^i cin 2046  (/)c0 2280

This theorem is referenced by:  inf3lem3 4615

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-reg 4593

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-in 2051  df-nul 2281

Copyright terms: Public domain