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Theorem ssini 3372
Description: An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)
Hypotheses
Ref Expression
ssini.1  |-  A  C_  B
ssini.2  |-  A  C_  C
Assertion
Ref Expression
ssini  |-  A  C_  ( B  i^i  C )

Proof of Theorem ssini
StepHypRef Expression
1 ssini.1 . . 3  |-  A  C_  B
2 ssini.2 . . 3  |-  A  C_  C
31, 2pm3.2i 272 . 2  |-  ( A 
C_  B  /\  A  C_  C )
4 ssin 3371 . 2  |-  ( ( A  C_  B  /\  A  C_  C )  <->  A  C_  ( B  i^i  C ) )
53, 4mpbi 145 1  |-  A  C_  ( B  i^i  C )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    i^i cin 3142    C_ wss 3143
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-v 2753  df-in 3149  df-ss 3156
This theorem is referenced by:  inv1  3473
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