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Theorem unielrel 5387
 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 4971 . 2
2 simpr 448 . 2
3 vex 2952 . . . . . 6
4 vex 2952 . . . . . 6
53, 4uniopel 4453 . . . . 5
65a1i 11 . . . 4
7 eleq1 2496 . . . 4
8 unieq 4017 . . . . 5
98eleq1d 2502 . . . 4
106, 7, 93imtr4d 260 . . 3
1110exlimivv 1645 . 2
121, 2, 11sylc 58 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wex 1550   wceq 1652   wcel 1725  cop 3810  cuni 4008   wrel 4876 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rex 2704  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-opab 4260  df-xp 4877  df-rel 4878
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