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Theorem inss 3727
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 3726 . 2  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  C )
2 incom 3691 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
3 ssinss1 3726 . . 3  |-  ( B 
C_  C  ->  ( B  i^i  A )  C_  C )
42, 3syl5eqss 3548 . 2  |-  ( B 
C_  C  ->  ( A  i^i  B )  C_  C )
51, 4jaoi 379 1  |-  ( ( A  C_  C  \/  B  C_  C )  -> 
( A  i^i  B
)  C_  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    i^i cin 3475    C_ wss 3476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-in 3483  df-ss 3490
This theorem is referenced by:  pmatcoe1fsupp  18966  ppttop  19271  inindif  27085  icccncfext  31226
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