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Theorem unss1 3516
Description: Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
unss1  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )

Proof of Theorem unss1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssel 3342 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21orim1d 813 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  \/  x  e.  C
)  ->  ( x  e.  B  \/  x  e.  C ) ) )
3 elun 3488 . . 3  |-  ( x  e.  ( A  u.  C )  <->  ( x  e.  A  \/  x  e.  C ) )
4 elun 3488 . . 3  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
52, 3, 43imtr4g 262 . 2  |-  ( A 
C_  B  ->  (
x  e.  ( A  u.  C )  ->  x  e.  ( B  u.  C ) ) )
65ssrdv 3354 1  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    e. wcel 1725    u. cun 3318    C_ wss 3320
This theorem is referenced by:  unss2  3518  unss12  3519  eldifpw  4755  tposss  6480  dftpos4  6498  hashbclem  11701  incexclem  12616  mreexexlem2d  13870  catcoppccl  14263  neitr  17244  restntr  17246  leordtval2  17276  cmpcld  17465  uniioombllem3  19477  limcres  19773  plyss  20118  shlej1  22862  orderseqlem  25527  pclfinclN  30747
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325  df-in 3327  df-ss 3334
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