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Theorem ssini 3556
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 3555 . 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 3311    C_ wss 3312
This theorem is referenced by:  inv1  3646  hartogslem1  7503  fbasrn  17908  limciun  19773  hlimcaui  22731  chdmm1i  22971  chm0i  22984  ledii  23030  lejdii  23032  mayetes3i  23224  mdslj2i  23815  mdslmd2i  23825  sumdmdlem2  23914  measunl  24562  ssoninhaus  26190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-ss 3326
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