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Theorem ssrin 3398
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 3187 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21anim1d 336 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  x  e.  C
)  ->  ( x  e.  B  /\  x  e.  C ) ) )
3 elin 3356 . . 3  |-  ( x  e.  ( A  i^i  C )  <->  ( x  e.  A  /\  x  e.  C ) )
4 elin 3356 . . 3  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
52, 3, 43imtr4g 205 . 2  |-  ( A 
C_  B  ->  (
x  e.  ( A  i^i  C )  ->  x  e.  ( B  i^i  C ) ) )
65ssrdv 3199 1  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2176    i^i cin 3165    C_ wss 3166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187
This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179
This theorem is referenced by:  sslin  3399  ssrind  3400  ss2in  3401  ssdisj  3517  ssdifin0  3542  ssres  4985  phplem2  6950  sbthlem7  7065  fiss  7079  tgss  14535  metrest  14978
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