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

Theorem elintg 3870
Description: Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)
Assertion
Ref Expression
elintg  |-  ( A  e.  V  ->  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem elintg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eleq1 2343 . 2  |-  ( y  =  A  ->  (
y  e.  |^| B  <->  A  e.  |^| B ) )
2 eleq1 2343 . . 3  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
32ralbidv 2563 . 2  |-  ( y  =  A  ->  ( A. x  e.  B  y  e.  x  <->  A. x  e.  B  A  e.  x ) )
4 vex 2791 . . 3  |-  y  e. 
_V
54elint2 3869 . 2  |-  ( y  e.  |^| B  <->  A. x  e.  B  y  e.  x )
61, 3, 5vtoclbg 2844 1  |-  ( A  e.  V  ->  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   A.wral 2543   |^|cint 3862
This theorem is referenced by:  elinti  3871  elrint  3903  onmindif  4482  onmindif2  4603  mremre  13506  toponmre  16830  1stcfb  17171  uffixfr  17618  plycpn  19669  insiga  23498  dfon2lem8  24146  intfmu2  25519  prcnt  25551  trintALTVD  28656  trintALT  28657
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
  Copyright terms: Public domain W3C validator