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Theorem inss 3570
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 3569 . 2  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  C )
2 incom 3533 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
3 ssinss1 3569 . . 3  |-  ( B 
C_  C  ->  ( B  i^i  A )  C_  C )
42, 3syl5eqss 3392 . 2  |-  ( B 
C_  C  ->  ( A  i^i  B )  C_  C )
51, 4jaoi 369 1  |-  ( ( A  C_  C  \/  B  C_  C )  -> 
( A  i^i  B
)  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    i^i cin 3319    C_ wss 3320
This theorem is referenced by:  ppttop  17071
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-in 3327  df-ss 3334
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