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Theorem ssun 3528
Description: A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
Assertion
Ref Expression
ssun  |-  ( ( A  C_  B  \/  A  C_  C )  ->  A  C_  ( B  u.  C ) )

Proof of Theorem ssun
StepHypRef Expression
1 ssun3 3514 . 2  |-  ( A 
C_  B  ->  A  C_  ( B  u.  C
) )
2 ssun4 3515 . 2  |-  ( A 
C_  C  ->  A  C_  ( B  u.  C
) )
31, 2jaoi 370 1  |-  ( ( A  C_  B  \/  A  C_  C )  ->  A  C_  ( B  u.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    u. cun 3320    C_ wss 3322
This theorem is referenced by:  pwunss  4490  pwssun  4491  ordssun  4683
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  df-in 3329  df-ss 3336
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