Theorem zfreg2 5699

Description: The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg 5698) 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 5697	. 2 |- (E.x x e. A -> E.x e. A A.y e. x -. y e. A)
3		n0 2884	. 2 |- (A =/= (/) <-> E.x x e. A)
4		incom 2787	. . . . 5 |- (A i^i x) = (x i^i A)
5	4	eqeq1i 1891	. . . 4 |- ((A i^i x) = (/) <-> (x i^i A) = (/))
6		disj 2914	. . . 4 |- ((x i^i A) = (/) <-> A.y e. x -. y e. A)
7	5, 6	bitri 190	. . 3 |- ((A i^i x) = (/) <-> A.y e. x -. y e. A)
8	7	rexbii 2128	. 2 |- (E.x e. A (A i^i x) = (/) <-> E.x e. A A.y e. x -. y e. A)
9	2, 3, 8	3imtr4i 236	1 |- (A =/= (/) -> E.x e. A (A i^i x) = (/))

Syntax hints:  -. wn 2   -> wi 3   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   i^i cin 2592  (/)c0 2875

This theorem is referenced by:  zfregfr 5706  zfregs 5754

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-reg 5695

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-in 2603  df-nul 2876

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