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Theorem inss 3218
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 3217 . 2  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  C )
2 incom 3181 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
3 ssinss1 3217 . . 3  |-  ( B 
C_  C  ->  ( B  i^i  A )  C_  C )
42, 3syl5eqss 3059 . 2  |-  ( B 
C_  C  ->  ( A  i^i  B )  C_  C )
51, 4jaoi 669 1  |-  ( ( A  C_  C  \/  B  C_  C )  -> 
( A  i^i  B
)  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 662    i^i cin 2987    C_ wss 2988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-in 2994  df-ss 3001
This theorem is referenced by: (None)
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