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Theorem ssinss1 3217
Description: Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)
Assertion
Ref Expression
ssinss1  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  C )

Proof of Theorem ssinss1
StepHypRef Expression
1 inss1 3209 . 2  |-  ( A  i^i  B )  C_  A
2 sstr2 3021 . 2  |-  ( ( A  i^i  B ) 
C_  A  ->  ( A  C_  C  ->  ( A  i^i  B )  C_  C ) )
31, 2ax-mp 7 1  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    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:  inss  3218
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