Theorem zfpair2 4692

Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 4691. See zfpair 4689 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.)

Assertion

Ref	Expression
zfpair2

Proof of Theorem zfpair2

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-pr 4691	. . . 4
2	1	bm1.3ii 4576	. . 3
3		dfcleq 2450	. . . . 5
4		vex 3112	. . . . . . . 8
5	4	elpr 4047	. . . . . . 7
6	5	bibi2i 313	. . . . . 6
7	6	albii 1640	. . . . 5
8	3, 7	bitri 249	. . . 4
9	8	exbii 1667	. . 3
10	2, 9	mpbir 209	. 2
11	10	issetri 3116	1

Syntax hints:  <->wb 184  \/wo 368  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  cvv 3109  {cpr 4031

This theorem is referenced by:  snex  4693  prex  4694  pwssun  4791  xpsspwOLD  5122  funopg  5625  fiint  7817  brdom7disj  8930  brdom6disj  8931  wlkntrllem1  24561  frisusgranb  24997  2pthfrgrarn  25009

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-sep 4573  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-v 3111  df-un 3480  df-sn 4030  df-pr 4032