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Theorem unss12 3319
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 3316 . 2  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )
2 unss2 3318 . 2  |-  ( C 
C_  D  ->  ( B  u.  C )  C_  ( B  u.  D
) )
31, 2sylan9ss 3180 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 104    u. cun 3139    C_ wss 3141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154
This theorem is referenced by:  ifssun  3560  fun  5400  resasplitss  5407  lspun  13554
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