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Theorem unielrel 5537
 Description: The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
unielrel

Proof of Theorem unielrel
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elrel 5110 . 2
2 simpr 461 . 2
3 vex 3112 . . . . . 6
4 vex 3112 . . . . . 6
53, 4uniopel 4756 . . . . 5
65a1i 11 . . . 4
7 eleq1 2529 . . . 4
8 unieq 4257 . . . . 5
98eleq1d 2526 . . . 4
106, 7, 93imtr4d 268 . . 3
1110exlimivv 1723 . 2
121, 2, 11sylc 60 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  <.cop 4035  U.cuni 4249  Relwrel 5009 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-rex 2813  df-v 3111  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-opab 4511  df-xp 5010  df-rel 5011
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