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Theorem unielrel 4869
 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 4462 . 2
2 simpr 108 . 2
3 vex 2605 . . . . . 6
4 vex 2605 . . . . . 6
53, 4uniopel 4013 . . . . 5
65a1i 9 . . . 4
7 eleq1 2142 . . . 4
8 unieq 3612 . . . . 5
98eleq1d 2148 . . . 4
106, 7, 93imtr4d 201 . . 3
1110exlimivv 1818 . 2
121, 2, 11sylc 61 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wceq 1285  wex 1422   wcel 1434  cop 3403  cuni 3603   wrel 4370 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-opab 3842  df-xp 4371  df-rel 4372 This theorem is referenced by: (None)
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