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Theorem disjel 3995
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 3993 . . 3 ((𝐴𝐵) = ∅ ↔ 𝐴 = (𝐴𝐵))
2 eleq2 2687 . . . 4 (𝐴 = (𝐴𝐵) → (𝐶𝐴𝐶 ∈ (𝐴𝐵)))
3 eldifn 3711 . . . 4 (𝐶 ∈ (𝐴𝐵) → ¬ 𝐶𝐵)
42, 3syl6bi 243 . . 3 (𝐴 = (𝐴𝐵) → (𝐶𝐴 → ¬ 𝐶𝐵))
51, 4sylbi 207 . 2 ((𝐴𝐵) = ∅ → (𝐶𝐴 → ¬ 𝐶𝐵))
65imp 445 1 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴) → ¬ 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  cdif 3552  cin 3554  c0 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3188  df-dif 3558  df-in 3562  df-nul 3892
This theorem is referenced by:  disjxun  4611  fvun1  6226  dedekindle  10145  fprodsplit  14621  unelldsys  29999  dvasin  33125
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