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Theorem uneqri 3082
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 3081 . . 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 173 . 2  |-  ( x  e.  ( A  u.  B )  <->  x  e.  C )
43eqriv 2037 1  |-  ( A  u.  B )  =  C
Colors of variables: wff set class
Syntax hints:    <-> wb 98    \/ wo 629    = wceq 1243    e. wcel 1393    u. cun 2912
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-un 2919
This theorem is referenced by:  unidm  3083  uncom  3084  unass  3097  undi  3182  unab  3201  un0  3248
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