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Theorem eluniab 3583
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  U. {  |  }
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem eluniab
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eluni 3574 . 2  U. {  |  }  {  |  }
2 nfv 1418 . . . 4  F/
3 nfsab1 2027 . . . 4  F/  {  |  }
42, 3nfan 1454 . . 3  F/  {  |  }
5 nfv 1418 . . 3  F/
6 eleq2 2098 . . . 4
7 eleq1 2097 . . . . 5  {  |  } 
{  |  }
8 abid 2025 . . . . 5  {  |  }
97, 8syl6bb 185 . . . 4  {  |  }
106, 9anbi12d 442 . . 3  {  |  }
114, 5, 10cbvex 1636 . 2  {  |  }
121, 11bitri 173 1  U. {  |  }
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98  wex 1378   wcel 1390   {cab 2023   U.cuni 3571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-uni 3572
This theorem is referenced by:  elunirab  3584  dfiun2g  3680  inuni  3900  snnex  4147  elfv  5119  unielxp  5742  tfrlem9  5876  tfr0  5878
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