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Theorem bnj1138 29221
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1138.1  |-  A  =  ( B  u.  C
)
Assertion
Ref Expression
bnj1138  |-  ( X  e.  A  <->  ( X  e.  B  \/  X  e.  C ) )

Proof of Theorem bnj1138
StepHypRef Expression
1 bnj1138.1 . . 3  |-  A  =  ( B  u.  C
)
21eleq2i 2502 . 2  |-  ( X  e.  A  <->  X  e.  ( B  u.  C
) )
3 elun 3490 . 2  |-  ( X  e.  ( B  u.  C )  <->  ( X  e.  B  \/  X  e.  C ) )
42, 3bitri 242 1  |-  ( X  e.  A  <->  ( X  e.  B  \/  X  e.  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    \/ wo 359    = wceq 1653    e. wcel 1726    u. cun 3320
This theorem is referenced by:  bnj1424  29272  bnj1408  29467  bnj1417  29472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327
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