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Theorem unss1 3520
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 3345 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21orim1d 835 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  \/  x  e.  C
)  ->  ( x  e.  B  \/  x  e.  C ) ) )
3 elun 3492 . . 3  |-  ( x  e.  ( A  u.  C )  <->  ( x  e.  A  \/  x  e.  C ) )
4 elun 3492 . . 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 3357 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 1756    u. cun 3321    C_ wss 3323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2969  df-un 3328  df-in 3330  df-ss 3337
This theorem is referenced by:  unss2  3522  unss12  3523  eldifpw  6383  tposss  6741  dftpos4  6759  hashbclem  12197  incexclem  13291  mreexexlem2d  14575  catcoppccl  14968  neitr  18759  restntr  18761  leordtval2  18791  cmpcld  18980  uniioombllem3  21040  limcres  21336  plyss  21642  shlej1  24714  orderseqlem  27664  bj-rrhatsscchat  32406  pclfinclN  33434
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