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Theorem ssun 3223
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 3209 . 2  |-  ( A 
C_  B  ->  A  C_  ( B  u.  C
) )
2 ssun4 3210 . 2  |-  ( A 
C_  C  ->  A  C_  ( B  u.  C
) )
31, 2jaoi 688 1  |-  ( ( A  C_  B  \/  A  C_  C )  ->  A  C_  ( B  u.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 680    u. cun 3037    C_ wss 3039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052
This theorem is referenced by:  pwunss  4173  pwssunim  4174
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