Theorem axpr 4690

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 4691 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

Distinct variable groups:   ,,   ,,

Proof of Theorem axpr

Step	Hyp	Ref	Expression
1		zfpair 4689	. . 3
2	1	isseti 3115	. 2
3		dfcleq 2450	. . 3
4		vex 3112	. . . . . . 7
5	4	elpr 4047	. . . . . 6
6	5	bibi2i 313	. . . . 5
7		bi2 198	. . . . 5
8	6, 7	sylbi 195	. . . 4
9	8	alimi 1633	. . 3
10	3, 9	sylbi 195	. 2
11	2, 10	eximii 1658	1

Syntax hints:  ->wi 4  <->wb 184  \/wo 368  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  {cpr 4031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-pw 4014  df-sn 4030  df-pr 4032