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Theorem unss1 3678
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 3503 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21orim1d 837 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  \/  x  e.  C
)  ->  ( x  e.  B  \/  x  e.  C ) ) )
3 elun 3650 . . 3  |-  ( x  e.  ( A  u.  C )  <->  ( x  e.  A  \/  x  e.  C ) )
4 elun 3650 . . 3  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
52, 3, 43imtr4g 270 . 2  |-  ( A 
C_  B  ->  (
x  e.  ( A  u.  C )  ->  x  e.  ( B  u.  C ) ) )
65ssrdv 3515 1  |-  ( A 
C_  B  ->  ( A  u.  C )  C_  ( B  u.  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    e. wcel 1767    u. cun 3479    C_ wss 3481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-un 3486  df-in 3488  df-ss 3495
This theorem is referenced by:  unss2  3680  unss12  3681  eldifpw  6607  tposss  6968  dftpos4  6986  hashbclem  12482  incexclem  13628  mreexexlem2d  14917  catcoppccl  15310  neitr  19549  restntr  19551  leordtval2  19581  cmpcld  19770  uniioombllem3  21862  limcres  22158  plyss  22464  shlej1  26092  ss2mcls  28744  orderseqlem  29250  bj-rrhatsscchat  34067  pclfinclN  35102
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