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Theorem inelcm 3387
Description: The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
Assertion
Ref Expression
inelcm ((𝐴𝐵𝐴𝐶) → (𝐵𝐶) ≠ ∅)

Proof of Theorem inelcm
StepHypRef Expression
1 elin 3223 . 2 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
2 ne0i 3333 . 2 (𝐴 ∈ (𝐵𝐶) → (𝐵𝐶) ≠ ∅)
31, 2sylbir 134 1 ((𝐴𝐵𝐴𝐶) → (𝐵𝐶) ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1461  wne 2280  cin 3034  c0 3327
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 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-v 2657  df-dif 3037  df-in 3041  df-nul 3328
This theorem is referenced by:  minel  3388  disjiun  3888
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