ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  inelcm GIF version

Theorem inelcm 3483
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 3318 . 2 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
2 ne0i 3429 . 2 (𝐴 ∈ (𝐵𝐶) → (𝐵𝐶) ≠ ∅)
31, 2sylbir 135 1 ((𝐴𝐵𝐴𝐶) → (𝐵𝐶) ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2148  wne 2347  cin 3128  c0 3422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-v 2739  df-dif 3131  df-in 3135  df-nul 3423
This theorem is referenced by:  minel  3484  disjiun  3995
  Copyright terms: Public domain W3C validator