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Theorem uneqri 3614
Description: Inference from membership to union. (Contributed by NM, 21-Jun-1993.)
Hypothesis
Ref Expression
uneqri.1  |-  ( ( x  e.  A  \/  x  e.  B )  <->  x  e.  C )
Assertion
Ref Expression
uneqri  |-  ( A  u.  B )  =  C
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem uneqri
StepHypRef Expression
1 elun 3612 . . 3  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
2 uneqri.1 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  <->  x  e.  C )
31, 2bitri 252 . 2  |-  ( x  e.  ( A  u.  B )  <->  x  e.  C )
43eqriv 2425 1  |-  ( A  u.  B )  =  C
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    \/ wo 369    = wceq 1437    e. wcel 1870    u. cun 3440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089  df-un 3447
This theorem is referenced by:  unidm  3615  uncom  3616  unass  3629  dfun2  3714  undi  3726  unab  3746  un0  3793  inundif  3879
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