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Theorem inelcm 3362
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 3198 . 2  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
2 ne0i 3308 . 2  |-  ( A  e.  ( B  i^i  C )  ->  ( B  i^i  C )  =/=  (/) )
31, 2sylbir 134 1  |-  ( ( A  e.  B  /\  A  e.  C )  ->  ( B  i^i  C
)  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1445    =/= wne 2262    i^i cin 3012   (/)c0 3302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-v 2635  df-dif 3015  df-in 3019  df-nul 3303
This theorem is referenced by:  minel  3363  disjiun  3862
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