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Theorem inss 3398
Description: Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)
Assertion
Ref Expression
inss  |-  ( ( A  C_  C  \/  B  C_  C )  -> 
( A  i^i  B
)  C_  C )

Proof of Theorem inss
StepHypRef Expression
1 ssinss1 3397 . 2  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  C )
2 incom 3361 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
3 ssinss1 3397 . . 3  |-  ( B 
C_  C  ->  ( B  i^i  A )  C_  C )
42, 3syl5eqss 3222 . 2  |-  ( B 
C_  C  ->  ( A  i^i  B )  C_  C )
51, 4jaoi 368 1  |-  ( ( A  C_  C  \/  B  C_  C )  -> 
( A  i^i  B
)  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    i^i cin 3151    C_ wss 3152
This theorem is referenced by:  ppttop  16744  pnfneige0  23374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166
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