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Theorem disjel 4409
 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 4406 . . 3 ((𝐴𝐵) = ∅ ↔ 𝐴 = (𝐴𝐵))
2 eleq2 2906 . . . 4 (𝐴 = (𝐴𝐵) → (𝐶𝐴𝐶 ∈ (𝐴𝐵)))
3 eldifn 4108 . . . 4 (𝐶 ∈ (𝐴𝐵) → ¬ 𝐶𝐵)
42, 3syl6bi 254 . . 3 (𝐴 = (𝐴𝐵) → (𝐶𝐴 → ¬ 𝐶𝐵))
51, 4sylbi 218 . 2 ((𝐴𝐵) = ∅ → (𝐶𝐴 → ¬ 𝐶𝐵))
65imp 407 1 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴) → ¬ 𝐶𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1530   ∈ wcel 2107   ∖ cdif 3937   ∩ cin 3939  ∅c0 4295 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-v 3502  df-dif 3943  df-in 3947  df-nul 4296 This theorem is referenced by:  disjxun  5061  fvun1  6753  dedekindle  10798  fprodsplit  15315  unelldsys  31322  dvasin  34864
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