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Theorem minelOLD 3986
 Description: Obsolete proof of minel 3985 as of 14-Jul-2021. (Contributed by NM, 22-Jun-1994.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
minelOLD ((𝐴𝐵 ∧ (𝐶𝐵) = ∅) → ¬ 𝐴𝐶)

Proof of Theorem minelOLD
StepHypRef Expression
1 inelcm 3984 . . . . 5 ((𝐴𝐶𝐴𝐵) → (𝐶𝐵) ≠ ∅)
21necon2bi 2812 . . . 4 ((𝐶𝐵) = ∅ → ¬ (𝐴𝐶𝐴𝐵))
3 imnan 437 . . . 4 ((𝐴𝐶 → ¬ 𝐴𝐵) ↔ ¬ (𝐴𝐶𝐴𝐵))
42, 3sylibr 223 . . 3 ((𝐶𝐵) = ∅ → (𝐴𝐶 → ¬ 𝐴𝐵))
54con2d 128 . 2 ((𝐶𝐵) = ∅ → (𝐴𝐵 → ¬ 𝐴𝐶))
65impcom 445 1 ((𝐴𝐵 ∧ (𝐶𝐵) = ∅) → ¬ 𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∩ cin 3539  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-in 3547  df-nul 3875 This theorem is referenced by: (None)
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