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Theorem elinti 3748
Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
elinti  |-  ( A  e.  |^| B  ->  ( C  e.  B  ->  A  e.  C ) )

Proof of Theorem elinti
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elintg 3747 . . 3  |-  ( A  e.  |^| B  ->  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x ) )
2 eleq2 2179 . . . 4  |-  ( x  =  C  ->  ( A  e.  x  <->  A  e.  C ) )
32rspccv 2758 . . 3  |-  ( A. x  e.  B  A  e.  x  ->  ( C  e.  B  ->  A  e.  C ) )
41, 3syl6bi 162 . 2  |-  ( A  e.  |^| B  ->  ( A  e.  |^| B  -> 
( C  e.  B  ->  A  e.  C ) ) )
54pm2.43i 49 1  |-  ( A  e.  |^| B  ->  ( C  e.  B  ->  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1463   A.wral 2391   |^|cint 3739
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-int 3740
This theorem is referenced by: (None)
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