Theorem zfpair2 4323

Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 4322. (Contributed by NM, 14-Nov-2006.)

Assertion

Ref	Expression
zfpair2	|- { x , y } e. _V

Proof of Theorem zfpair2

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-pr 4322	. . . 4 |- E. z A. w ( ( w = x \/ w = y ) -> w e. z )
2	1	bm1.3ii 4231	. . 3 |- E. z A. w ( w e. z <-> ( w = x \/ w = y ) )
3		dfcleq 2226	. . . . 5 |- ( z = { x , y } <-> A. w ( w e. z <-> w e. { x , y } ) )
4		vex 2816	. . . . . . . 8 |- w e. _V
5	4	elpr 3710	. . . . . . 7 |- ( w e. { x , y } <-> ( w = x \/ w = y ) )
6	5	bibi2i 227	. . . . . 6 |- ( ( w e. z <-> w e. { x , y } ) <-> ( w e. z <-> ( w = x \/ w = y ) ) )
7	6	albii 1519	. . . . 5 |- ( A. w ( w e. z <-> w e. { x , y } ) <-> A. w ( w e. z <-> ( w = x \/ w = y ) ) )
8	3, 7	bitri 184	. . . 4 |- ( z = { x , y } <-> A. w ( w e. z <-> ( w = x \/ w = y ) ) )
9	8	exbii 1654	. . 3 |- ( E. z z = { x , y } <-> E. z A. w ( w e. z <-> ( w = x \/ w = y ) ) )
10	2, 9	mpbir 146	. 2 |- E. z z = { x , y }
11	10	issetri 2823	1 |- { x , y } e. _V

Syntax hints:   <-> wb 105   \/ wo 716   A.wal 1396   = wceq 1398   E.wex 1541   e. wcel 2203   _Vcvv 2813   {cpr 3690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696

This theorem is referenced by:  vsnex  4324  prexg  4325  onintexmid  4695  funopg  5386  funopsn  5860  umgredg  16140