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Theorem eluniab 3748
 Description: Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
eluniab
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem eluniab
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eluni 3739 . 2
2 nfv 1508 . . . 4
3 nfsab1 2129 . . . 4
42, 3nfan 1544 . . 3
5 nfv 1508 . . 3
6 eleq2 2203 . . . 4
7 eleq1 2202 . . . . 5
8 abid 2127 . . . . 5
97, 8syl6bb 195 . . . 4
106, 9anbi12d 464 . . 3
114, 5, 10cbvex 1729 . 2
121, 11bitri 183 1
 Colors of variables: wff set class Syntax hints:   wa 103   wb 104  wex 1468   wcel 1480  cab 2125  cuni 3736 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-uni 3737 This theorem is referenced by:  elunirab  3749  dfiun2g  3845  inuni  4080  snnex  4369  elfv  5419  unielxp  6072  tfrlem9  6216  tfr0dm  6219  metrest  12685
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