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Theorem ss2in 3409
Description: Intersection of subclasses. (Contributed by NM, 5-May-2000.)
Assertion
Ref Expression
ss2in  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A  i^i  C
)  C_  ( B  i^i  D ) )

Proof of Theorem ss2in
StepHypRef Expression
1 ssrin 3407 . 2  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
2 sslin 3408 . 2  |-  ( C 
C_  D  ->  ( B  i^i  C )  C_  ( B  i^i  D ) )
31, 2sylan9ss 3205 1  |-  ( ( A  C_  B  /\  C  C_  D )  -> 
( A  i^i  C
)  C_  ( B  i^i  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    i^i cin 3164    C_ wss 3165
This theorem is referenced by:  disjxiun  4036  undom  6966  strlemor1  13251  strleun  13254  dprdss  15280  dprd2da  15293  ablfac1b  15321  tgcl  16723  innei  16878  hausnei2  17097  fbssfi  17548  fbunfip  17580  fgcl  17589  blin2  17991  5oai  22256  mayetes3i  22325  mdsl0  22906  vdgrun  23908  int2pre  25340  neibastop1  26411  heibor1lem  26636  pl42lem2N  30791  pl42lem3N  30792
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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