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Theorem ssinss1 3275
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 3266 . 2  |-  ( A  i^i  B )  C_  A
2 sstr2 3074 . 2  |-  ( ( A  i^i  B ) 
C_  A  ->  ( A  C_  C  ->  ( A  i^i  B )  C_  C ) )
31, 2ax-mp 5 1  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    i^i cin 3040    C_ wss 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054
This theorem is referenced by:  inss  3276  distop  12181  ntrin  12220  innei  12259  txcnp  12367
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