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Theorem inelcm 3881
Description: The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
Assertion
Ref Expression
inelcm  |-  ( ( A  e.  B  /\  A  e.  C )  ->  ( B  i^i  C
)  =/=  (/) )

Proof of Theorem inelcm
StepHypRef Expression
1 elin 3687 . 2  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
2 ne0i 3791 . 2  |-  ( A  e.  ( B  i^i  C )  ->  ( B  i^i  C )  =/=  (/) )
31, 2sylbir 213 1  |-  ( ( A  e.  B  /\  A  e.  C )  ->  ( B  i^i  C
)  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767    =/= wne 2662    i^i cin 3475   (/)c0 3785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-in 3483  df-nul 3786
This theorem is referenced by:  minel  3882  disji  4435  disjiun  4437  onnseq  7012  uniinqs  7388  en3lplem1  8027  cplem1  8303  fpwwe2lem12  9015  swrdn0  12612  limsupgre  13260  lmcls  19566  concn  19690  iunconlem  19691  concompclo  19699  2ndcsep  19723  txcls  19837  pthaus  19871  qtopeu  19949  trfbas2  20076  filss  20086  zfbas  20129  fmfnfm  20191  tsmsfbas  20358  restmetu  20822  qdensere  21009  reperflem  21055  reconnlem1  21063  metds0  21086  metnrmlem1a  21094  minveclem3b  21575  ovolicc2lem5  21664  taylfval  22485  wwlkm1edg  24408  disjif  27109  disjif2  27112  subfacp1lem6  28266  erdszelem5  28276  pconcon  28313  cvmseu  28358  lfinpfin  29773  locfincmp  29774  neibastop2lem  29779  sstotbnd3  29873
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