Theorem zfpair2 4061

Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 4060. (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 4060	. . . 4 |- E. z A. w ( ( w = x \/ w = y ) -> w e. z )
2	1	bm1.3ii 3981	. . 3 |- E. z A. w ( w e. z <-> ( w = x \/ w = y ) )
3		dfcleq 2089	. . . . 5 |- ( z = { x , y } <-> A. w ( w e. z <-> w e. { x , y } ) )
4		vex 2636	. . . . . . . 8 |- w e. _V
5	4	elpr 3487	. . . . . . 7 |- ( w e. { x , y } <-> ( w = x \/ w = y ) )
6	5	bibi2i 226	. . . . . 6 |- ( ( w e. z <-> w e. { x , y } ) <-> ( w e. z <-> ( w = x \/ w = y ) ) )
7	6	albii 1411	. . . . 5 |- ( A. w ( w e. z <-> w e. { x , y } ) <-> A. w ( w e. z <-> ( w = x \/ w = y ) ) )
8	3, 7	bitri 183	. . . 4 |- ( z = { x , y } <-> A. w ( w e. z <-> ( w = x \/ w = y ) ) )
9	8	exbii 1548	. . 3 |- ( E. z z = { x , y } <-> E. z A. w ( w e. z <-> ( w = x \/ w = y ) ) )
10	2, 9	mpbir 145	. 2 |- E. z z = { x , y }
11	10	issetri 2642	1 |- { x , y } e. _V

Syntax hints:   <-> wb 104   \/ wo 667   A.wal 1294   = wceq 1296   E.wex 1433   e. wcel 1445   _Vcvv 2633   {cpr 3467

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pr 4060

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-un 3017  df-sn 3472  df-pr 3473

This theorem is referenced by:  prexg  4062  onintexmid  4416  funopg  5082