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Theorem ssini 3392
Description: An inference showing that the 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 441 . 2  |-  ( A 
C_  B  /\  A  C_  C )
4 ssin 3391 . 2  |-  ( ( A  C_  B  /\  A  C_  C )  <->  A  C_  ( B  i^i  C ) )
53, 4mpbi 199 1  |-  A  C_  ( B  i^i  C )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    i^i cin 3151    C_ wss 3152
This theorem is referenced by:  inv1  3481  hartogslem1  7257  fbasrn  17579  limciun  19244  hlimcaui  21816  chdmm1i  22056  chm0i  22069  ledii  22115  lejdii  22117  mayetes3i  22309  mdslj2i  22900  mdslmd2i  22910  sumdmdlem2  22999  ssoninhaus  24887  residcp  25077
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166
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