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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
StepHypRef Expression
1 zfpair 4689 . . 3
21isseti 3115 . 2
3 dfcleq 2450 . . 3
4 vex 3112 . . . . . . 7
54elpr 4047 . . . . . 6
65bibi2i 313 . . . . 5
7 bi2 198 . . . . 5
86, 7sylbi 195 . . . 4
98alimi 1633 . . 3
103, 9sylbi 195 . 2
112, 10eximii 1658 1
Colors of variables: wff setvar class
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
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