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Theorem opeliunxp2 5146
Description: Membership in a union of Cartesian products. (Contributed by Mario Carneiro, 14-Feb-2015.)
Hypothesis
Ref Expression
opeliunxp2.1
Assertion
Ref Expression
opeliunxp2
Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem opeliunxp2
StepHypRef Expression
1 df-br 4453 . . 3
2 relxp 5115 . . . . . 6
32rgenw 2818 . . . . 5
4 reliun 5128 . . . . 5
53, 4mpbir 209 . . . 4
65brrelexi 5045 . . 3
71, 6sylbir 213 . 2
8 elex 3118 . . 3
98adantr 465 . 2
10 nfiu1 4360 . . . . 5
1110nfel2 2637 . . . 4
12 nfv 1707 . . . 4
1311, 12nfbi 1934 . . 3
14 opeq1 4217 . . . . 5
1514eleq1d 2526 . . . 4
16 eleq1 2529 . . . . 5
17 opeliunxp2.1 . . . . . 6
1817eleq2d 2527 . . . . 5
1916, 18anbi12d 710 . . . 4
2015, 19bibi12d 321 . . 3
21 opeliunxp 5056 . . 3
2213, 20, 21vtoclg1f 3166 . 2
237, 9, 22pm5.21nii 353 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807   cvv 3109  {csn 4029  <.cop 4035  U_ciun 4330   class class class wbr 4452  X.cxp 5002  Relwrel 5009
This theorem is referenced by:  mpt2xopn0yelv  6960  mpt2xopxnop0  6962  eldmcoa  15392  dmdprd  17029  ply1frcl  18355  eldv  22302  perfdvf  22307  eltayl  22755  dfcnv2  27517  cvmliftlem1  28730  filnetlem3  30198
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-nul 4581  ax-pr 4691
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-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  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-iun 4332  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011
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