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Theorem ssini 3500
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 442 . 2  |-  ( A 
C_  B  /\  A  C_  C )
4 ssin 3499 . 2  |-  ( ( A  C_  B  /\  A  C_  C )  <->  A  C_  ( B  i^i  C ) )
53, 4mpbi 200 1  |-  A  C_  ( B  i^i  C )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    i^i cin 3255    C_ wss 3256
This theorem is referenced by:  inv1  3590  hartogslem1  7437  fbasrn  17830  limciun  19641  hlimcaui  22580  chdmm1i  22820  chm0i  22833  ledii  22879  lejdii  22881  mayetes3i  23073  mdslj2i  23664  mdslmd2i  23674  sumdmdlem2  23763  measunl  24357  ssoninhaus  25905
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894  df-in 3263  df-ss 3270
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