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Theorem ssrin 3223
Description: Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ssrin  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )

Proof of Theorem ssrin
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssel 3017 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21anim1d 329 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  x  e.  C
)  ->  ( x  e.  B  /\  x  e.  C ) ) )
3 elin 3181 . . 3  |-  ( x  e.  ( A  i^i  C )  <->  ( x  e.  A  /\  x  e.  C ) )
4 elin 3181 . . 3  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
52, 3, 43imtr4g 203 . 2  |-  ( A 
C_  B  ->  (
x  e.  ( A  i^i  C )  ->  x  e.  ( B  i^i  C ) ) )
65ssrdv 3029 1  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1438    i^i cin 2996    C_ wss 2997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010
This theorem is referenced by:  sslin  3224  ss2in  3225  ssdisj  3336  ssdifin0  3360  ssres  4726  phplem2  6549  sbthlem7  6651
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