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Theorem elinti 4003
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 4002 . . 3  |-  ( A  e.  |^| B  ->  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x ) )
2 eleq2 2450 . . . 4  |-  ( x  =  C  ->  ( A  e.  x  <->  A  e.  C ) )
32rspccv 2994 . . 3  |-  ( A. x  e.  B  A  e.  x  ->  ( C  e.  B  ->  A  e.  C ) )
41, 3syl6bi 220 . 2  |-  ( A  e.  |^| B  ->  ( A  e.  |^| B  -> 
( C  e.  B  ->  A  e.  C ) ) )
54pm2.43i 45 1  |-  ( A  e.  |^| B  ->  ( C  e.  B  ->  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   A.wral 2651   |^|cint 3994
This theorem is referenced by:  inttsk  8584  subgint  14893  subrgint  15819  lssintcl  15969  ufinffr  17884  shintcli  22681  insiga  24318
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-v 2903  df-int 3995
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