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Theorem elinti 3871
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 3870 . . 3  |-  ( A  e.  |^| B  ->  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x ) )
2 eleq2 2344 . . . 4  |-  ( x  =  C  ->  ( A  e.  x  <->  A  e.  C ) )
32rspccv 2881 . . 3  |-  ( A. x  e.  B  A  e.  x  ->  ( C  e.  B  ->  A  e.  C ) )
41, 3syl6bi 219 . 2  |-  ( A  e.  |^| B  ->  ( A  e.  |^| B  -> 
( C  e.  B  ->  A  e.  C ) ) )
54pm2.43i 43 1  |-  ( A  e.  |^| B  ->  ( C  e.  B  ->  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   A.wral 2543   |^|cint 3862
This theorem is referenced by:  inttsk  8396  subgint  14641  subrgint  15567  lssintcl  15721  ufinffr  17624  shintcli  21908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-int 3863
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