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Theorem unss1 3460
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 3286 . . . 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 3432 . . 3  |-  ( x  e.  ( A  u.  C )  <->  ( x  e.  A  \/  x  e.  C ) )
4 elun 3432 . . 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 3298 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 1717    u. cun 3262    C_ wss 3264
This theorem is referenced by:  unss2  3462  unss12  3463  eldifpw  4696  tposss  6417  dftpos4  6435  hashbclem  11629  incexclem  12544  mreexexlem2d  13798  catcoppccl  14191  neitr  17167  restntr  17169  leordtval2  17199  cmpcld  17388  uniioombllem3  19345  limcres  19641  plyss  19986  shlej1  22711  orderseqlem  25277  pclfinclN  30065
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902  df-un 3269  df-in 3271  df-ss 3278
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