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Theorem ssini 3405
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 441 . 2  |-  ( A 
C_  B  /\  A  C_  C )
4 ssin 3404 . 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 3164    C_ wss 3165
This theorem is referenced by:  inv1  3494  hartogslem1  7273  fbasrn  17595  limciun  19260  hlimcaui  21832  chdmm1i  22072  chm0i  22085  ledii  22131  lejdii  22133  mayetes3i  22325  mdslj2i  22916  mdslmd2i  22926  sumdmdlem2  23015  ssoninhaus  24959  residcp  25180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179
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