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Theorem ss2in 3504
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 3502 . 2  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
2 sslin 3503 . 2  |-  ( C 
C_  D  ->  ( B  i^i  C )  C_  ( B  i^i  D ) )
31, 2sylan9ss 3297 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 359    i^i cin 3255    C_ wss 3256
This theorem is referenced by:  disjxiun  4143  undom  7125  strlemor1  13476  strleun  13479  dprdss  15507  dprd2da  15520  ablfac1b  15548  tgcl  16950  innei  17105  hausnei2  17332  fbssfi  17783  fbunfip  17815  fgcl  17824  blin2  18342  vdgrun  21513  vdgrfiun  21514  5oai  23004  mayetes3i  23073  mdsl0  23654  neibastop1  26072  heibor1lem  26202  pl42lem2N  30145  pl42lem3N  30146
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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