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Theorem disjel 3302
 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 3300 . . 3 ((𝐴𝐵) = ∅ ↔ 𝐴 = (𝐴𝐵))
2 eleq2 2117 . . . 4 (𝐴 = (𝐴𝐵) → (𝐶𝐴𝐶 ∈ (𝐴𝐵)))
3 eldifn 3095 . . . 4 (𝐶 ∈ (𝐴𝐵) → ¬ 𝐶𝐵)
42, 3syl6bi 156 . . 3 (𝐴 = (𝐴𝐵) → (𝐶𝐴 → ¬ 𝐶𝐵))
51, 4sylbi 118 . 2 ((𝐴𝐵) = ∅ → (𝐶𝐴 → ¬ 𝐶𝐵))
65imp 119 1 (((𝐴𝐵) = ∅ ∧ 𝐶𝐴) → ¬ 𝐶𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   = wceq 1259   ∈ wcel 1409   ∖ cdif 2942   ∩ cin 2944  ∅c0 3252 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-dif 2948  df-in 2952  df-nul 3253 This theorem is referenced by:  fvun1  5267
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