MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elinti Unicode version

Theorem elinti 3887
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 3886 . . 3  |-  ( A  e.  |^| B  ->  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x ) )
2 eleq2 2357 . . . 4  |-  ( x  =  C  ->  ( A  e.  x  <->  A  e.  C ) )
32rspccv 2894 . . 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 1696   A.wral 2556   |^|cint 3878
This theorem is referenced by:  inttsk  8412  subgint  14657  subrgint  15583  lssintcl  15737  ufinffr  17640  shintcli  21924  insiga  23513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-int 3879
  Copyright terms: Public domain W3C validator