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Theorem uneqri 3449
Description: Inference from membership to union. (Contributed by NM, 5-Aug-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 3448 . . 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 241 . 2  |-  ( x  e.  ( A  u.  B )  <->  x  e.  C )
43eqriv 2401 1  |-  ( A  u.  B )  =  C
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    = wceq 1649    e. wcel 1721    u. cun 3278
This theorem is referenced by:  unidm  3450  uncom  3451  unass  3464  dfun2  3536  undi  3548  unab  3568  un0  3612  inundif  3666
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-un 3285
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