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

Theorem disjel 3448
Description: A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)
Assertion
Ref Expression
disjel (((𝐴𝐵) = ∅ ∧ 𝐶𝐴) → ¬ 𝐶𝐵)

Proof of Theorem disjel
StepHypRef Expression
1 disj3 3446 . . 3 ((𝐴𝐵) = ∅ ↔ 𝐴 = (𝐴𝐵))
2 eleq2 2221 . . . 4 (𝐴 = (𝐴𝐵) → (𝐶𝐴𝐶 ∈ (𝐴𝐵)))
3 eldifn 3230 . . . 4 (𝐶 ∈ (𝐴𝐵) → ¬ 𝐶𝐵)
42, 3syl6bi 162 . . 3 (𝐴 = (𝐴𝐵) → (𝐶𝐴 → ¬ 𝐶𝐵))
51, 4sylbi 120 . 2 ((𝐴𝐵) = ∅ → (𝐶𝐴 → ¬ 𝐶𝐵))
65imp 123 1 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴) → ¬ 𝐶𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1335  wcel 2128  cdif 3099  cin 3101  c0 3394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-dif 3104  df-in 3108  df-nul 3395
This theorem is referenced by:  fvun1  5535  ctssdccl  7056  fsumsplit  11308  fprodsplitdc  11497  fprodsplit  11498
  Copyright terms: Public domain W3C validator