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Theorem unss12 3360
Description: Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
Assertion
Ref Expression
unss12  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A  u.  C
)  C_  ( B  u.  D ) )

Proof of Theorem unss12
StepHypRef Expression
1 unss1 3357 . 2  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )
2 unss2 3359 . 2  |-  ( C 
C_  D  ->  ( B  u.  C )  C_  ( B  u.  D
) )
31, 2sylan9ss 3205 1  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A  u.  C
)  C_  ( B  u.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    u. cun 3163    C_ wss 3165
This theorem is referenced by:  pwssun  4315  fun  5421  undom  6966  finsschain  7178  dprd2da  15293  dmdprdsplit2lem  15296  lspun  15760  spanuni  22139  sshhococi  22141  trunitr  25212  mvdco  27491  dochdmj1  32202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-ss 3179
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