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Theorem unss1 3357
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 3187 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21orim1d 812 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  \/  x  e.  C
)  ->  ( x  e.  B  \/  x  e.  C ) ) )
3 elun 3329 . . 3  |-  ( x  e.  ( A  u.  C )  <->  ( x  e.  A  \/  x  e.  C ) )
4 elun 3329 . . 3  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
52, 3, 43imtr4g 261 . 2  |-  ( A 
C_  B  ->  (
x  e.  ( A  u.  C )  ->  x  e.  ( B  u.  C ) ) )
65ssrdv 3198 1  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    e. wcel 1696    u. cun 3163    C_ wss 3165
This theorem is referenced by:  unss2  3359  unss12  3360  eldifpw  4582  tposss  6251  dftpos4  6269  hashbclem  11406  incexclem  12311  mreexexlem2d  13563  catcoppccl  13956  restntr  16928  leordtval2  16958  cmpcld  17145  uniioombllem3  18956  limcres  19252  plyss  19597  shlej1  21955  orderseqlem  24323  pclfinclN  30761
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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