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Theorem unieq 3580
Description: Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
unieq  U.  U.

Proof of Theorem unieq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexeq 2500 . . 3
21abbidv 2152 . 2  {  |  }  {  |  }
3 dfuni2 3573 . 2  U.  {  |  }
4 dfuni2 3573 . 2  U.  {  |  }
52, 3, 43eqtr4g 2094 1  U.  U.
Colors of variables: wff set class
Syntax hints:   wi 4   wceq 1242   {cab 2023  wrex 2301   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-rex 2306  df-uni 3572
This theorem is referenced by:  unieqi  3581  unieqd  3582  uniintsnr  3642  iununir  3729  treq  3851  limeq  4080  uniex  4140  uniexg  4141  ordsucunielexmid  4216  elvvuni  4347  unielrel  4788  unixp0im  4797  iotass  4827  en1bg  6216  bj-uniex  9302  bj-uniexg  9303
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