Theorem zfpair2 4907

Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 4906. See zfpair 4904 for its derivation from the other axioms. (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 4906	. . . 4 |- E. z A. w ( ( w = x \/ w = y ) -> w e. z )
2	1	bm1.3ii 4784	. . 3 |- E. z A. w ( w e. z <-> ( w = x \/ w = y ) )
3		dfcleq 2616	. . . . 5 |- ( z = { x , y } <-> A. w ( w e. z <-> w e. { x , y } ) )
4		vex 3203	. . . . . . . 8 |- w e. _V
5	4	elpr 4198	. . . . . . 7 |- ( w e. { x , y } <-> ( w = x \/ w = y ) )
6	5	bibi2i 327	. . . . . 6 |- ( ( w e. z <-> w e. { x , y } ) <-> ( w e. z <-> ( w = x \/ w = y ) ) )
7	6	albii 1747	. . . . 5 |- ( A. w ( w e. z <-> w e. { x , y } ) <-> A. w ( w e. z <-> ( w = x \/ w = y ) ) )
8	3, 7	bitri 264	. . . 4 |- ( z = { x , y } <-> A. w ( w e. z <-> ( w = x \/ w = y ) ) )
9	8	exbii 1774	. . 3 |- ( E. z z = { x , y } <-> E. z A. w ( w e. z <-> ( w = x \/ w = y ) ) )
10	2, 9	mpbir 221	. 2 |- E. z z = { x , y }
11	10	issetri 3210	1 |- { x , y } e. _V

Syntax hints:   <-> wb 196   \/ wo 383   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  snex  4908  prex  4909  pwssun  5020  xpsspw  5233  funopg  5922  fiint  8237  brdom7disj  9353  brdom6disj  9354  2pthfrgrrn  27146  sprval  41729