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Theorem ss2in 3164
Description: Intersection of subclasses. (Contributed by NM, 5-May-2000.)
Assertion
Ref Expression
ss2in  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A  i^i  C
)  C_  ( B  i^i  D ) )

Proof of Theorem ss2in
StepHypRef Expression
1 ssrin 3162 . 2  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
2 sslin 3163 . 2  |-  ( C 
C_  D  ->  ( B  i^i  C )  C_  ( B  i^i  D ) )
31, 2sylan9ss 2958 1  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A  i^i  C
)  C_  ( B  i^i  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    i^i cin 2916    C_ wss 2917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931
This theorem is referenced by: (None)
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