Theorem zfreg2 4584

Description: The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg 4583) is formally more convenient for us in some cases. Axiom Reg of [BellMachover] p. 480.

Hypothesis

Ref	Expression
zfreg2.1	|- A e. V

Assertion

Ref	Expression
zfreg2	|- (A =/= (/) -> E.x e. A (A i^i x) = (/))

Distinct variable group:   x,A

Proof of Theorem zfreg2

Step	Hyp	Ref	Expression
1		zfreg2.1	. . 3 |- A e. V
2	1	zfregcl 4582	. 2 |- (E.x x e. A -> E.x e. A A.y e. x -. y e. A)
3		ne0 2286	. 2 |- (A =/= (/) <-> E.x x e. A)
4		incom 2206	. . . . 5 |- (A i^i x) = (x i^i A)
5	4	eqeq1i 1481	. . . 4 |- ((A i^i x) = (/) <-> (x i^i A) = (/))
6		disj 2309	. . . 4 |- ((x i^i A) = (/) <-> A.y e. x -. y e. A)
7	5, 6	bitr 173	. . 3 |- ((A i^i x) = (/) <-> A.y e. x -. y e. A)
8	7	rexbii 1667	. 2 |- (E.x e. A (A i^i x) = (/) <-> E.x e. A A.y e. x -. y e. A)
9	2, 3, 8	3imtr4 219	1 |- (A =/= (/) -> E.x e. A (A i^i x) = (/))

Syntax hints:  -. wn 2   -> wi 3   = wceq 955   e. wcel 957  E.wex 979   =/= wne 1584  A.wral 1644  E.wrex 1645  Vcvv 1809   i^i cin 2044  (/)c0 2278

This theorem is referenced by:  zfregfr 4588  zfregs 4634

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-reg 4580

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-dif 2047  df-in 2049  df-nul 2279

Copyright terms: Public domain