Theorem zfpair2 4300

Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 4299. (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 4299	. . . 4 |- E. z A. w ( ( w = x \/ w = y ) -> w e. z )
2	1	bm1.3ii 4210	. . 3 |- E. z A. w ( w e. z <-> ( w = x \/ w = y ) )
3		dfcleq 2225	. . . . 5 |- ( z = { x , y } <-> A. w ( w e. z <-> w e. { x , y } ) )
4		vex 2805	. . . . . . . 8 |- w e. _V
5	4	elpr 3690	. . . . . . 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 1518	. . . . 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 1653	. . 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 2812	1 |- { x , y } e. _V

Syntax hints:   <-> wb 105   \/ wo 715   A.wal 1395   = wceq 1397   E.wex 1540   e. wcel 2202   _Vcvv 2802   {cpr 3670

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676

This theorem is referenced by:  prexg  4301  onintexmid  4671  funopg  5360  funopsn  5829  umgredg  15995