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Theorem uneqri 3646
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 3645 . . 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 249 . 2  |-  ( x  e.  ( A  u.  B )  <->  x  e.  C )
43eqriv 2463 1  |-  ( A  u.  B )  =  C
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    = wceq 1379    e. wcel 1767    u. cun 3474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-un 3481
This theorem is referenced by:  unidm  3647  uncom  3648  unass  3661  dfun2  3733  undi  3745  unab  3765  un0  3810  inundif  3905
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