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Theorem elopabi 6861
 Description: A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
Hypotheses
Ref Expression
elopabi.1
elopabi.2
Assertion
Ref Expression
elopabi
Distinct variable groups:   ,,   ,,

Proof of Theorem elopabi
StepHypRef Expression
1 relopab 5134 . . . 4
2 1st2nd 6846 . . . 4
31, 2mpan 670 . . 3
4 id 22 . . 3
53, 4eqeltrrd 2546 . 2
6 fvex 5881 . . 3
7 fvex 5881 . . 3
8 elopabi.1 . . 3
9 elopabi.2 . . 3
106, 7, 8, 9opelopab 4774 . 2
115, 10sylib 196 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  e.wcel 1818  <.cop 4035  {copab 4509  Relwrel 5009  `cfv 5593   c1st 6798   c2nd 6799 This theorem is referenced by:  uhgrac  24305  wlkelwrd  24530  drngoi  25409  vci  25441  dicelval1sta  36914 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-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592 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-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fv 5601  df-1st 6800  df-2nd 6801
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