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Theorem ssini 3685
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 456 . 2  |-  ( A 
C_  B  /\  A  C_  C )
4 ssin 3684 . 2  |-  ( ( A  C_  B  /\  A  C_  C )  <->  A  C_  ( B  i^i  C ) )
53, 4mpbi 211 1  |-  A  C_  ( B  i^i  C )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    i^i cin 3435    C_ wss 3436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-in 3443  df-ss 3450
This theorem is referenced by:  inv1  3789  hartogslem1  8060  xptrrel  13033  fbasrn  20886  limciun  22836  hlimcaui  26875  chdmm1i  27116  chm0i  27129  ledii  27175  lejdii  27177  mayetes3i  27368  mdslj2i  27959  mdslmd2i  27969  sumdmdlem2  28058  sigapildsys  28980  ssoninhaus  31101  fouriersw  37915  sge0split  38039
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