Theorem axpr 4905

Description: Unabbreviated version of the Axiom of Pairing of ZF set theory, derived as a theorem from the other axioms.

This theorem should not be referenced by any proof. Instead, use ax-pr 4906 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
axpr	|- E. z A. w ( ( w = x \/ w = y ) -> w e. z )

Distinct variable groups:   x, z, w   y, z, w

Proof of Theorem axpr

Step	Hyp	Ref	Expression
1		zfpair 4904	. . 3 |- { x , y } e. _V
2	1	isseti 3209	. 2 |- E. z z = { x , y }
3		dfcleq 2616	. . 3 |- ( z = { x , y } <-> A. w ( w e. z <-> w e. { x , y } ) )
4		vex 3203	. . . . . . 7 |- w e. _V
5	4	elpr 4198	. . . . . 6 |- ( w e. { x , y } <-> ( w = x \/ w = y ) )
6	5	bibi2i 327	. . . . 5 |- ( ( w e. z <-> w e. { x , y } ) <-> ( w e. z <-> ( w = x \/ w = y ) ) )
7		biimpr 210	. . . . 5 |- ( ( w e. z <-> ( w = x \/ w = y ) ) -> ( ( w = x \/ w = y ) -> w e. z ) )
8	6, 7	sylbi 207	. . . 4 |- ( ( w e. z <-> w e. { x , y } ) -> ( ( w = x \/ w = y ) -> w e. z ) )
9	8	alimi 1739	. . 3 |- ( A. w ( w e. z <-> w e. { x , y } ) -> A. w ( ( w = x \/ w = y ) -> w e. z ) )
10	3, 9	sylbi 207	. 2 |- ( z = { x , y } -> A. w ( ( w = x \/ w = y ) -> w e. z ) )
11	2, 10	eximii 1764	1 |- E. z A. w ( ( w = x \/ w = y ) -> w e. z )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180

This theorem is referenced by: (None)