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Theorem disjel 4223
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 4220 . . 3 ((𝐴𝐵) = ∅ ↔ 𝐴 = (𝐴𝐵))
2 eleq2 2871 . . . 4 (𝐴 = (𝐴𝐵) → (𝐶𝐴𝐶 ∈ (𝐴𝐵)))
3 eldifn 3935 . . . 4 (𝐶 ∈ (𝐴𝐵) → ¬ 𝐶𝐵)
42, 3syl6bi 245 . . 3 (𝐴 = (𝐴𝐵) → (𝐶𝐴 → ¬ 𝐶𝐵))
51, 4sylbi 209 . 2 ((𝐴𝐵) = ∅ → (𝐶𝐴 → ¬ 𝐶𝐵))
65imp 396 1 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴) → ¬ 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385   = wceq 1653  wcel 2157  cdif 3770  cin 3772  c0 4119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2781
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ral 3098  df-v 3391  df-dif 3776  df-in 3780  df-nul 4120
This theorem is referenced by:  disjxun  4845  fvun1  6498  dedekindle  10495  fprodsplit  15037  unelldsys  30741  dvasin  33988
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