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Theorem minelOLD 4067
Description: Obsolete proof of minel 4066 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 4065 . . . . 5 ((𝐴𝐶𝐴𝐵) → (𝐶𝐵) ≠ ∅)
21necon2bi 2853 . . . 4 ((𝐶𝐵) = ∅ → ¬ (𝐴𝐶𝐴𝐵))
3 imnan 437 . . . 4 ((𝐴𝐶 → ¬ 𝐴𝐵) ↔ ¬ (𝐴𝐶𝐴𝐵))
42, 3sylibr 224 . . 3 ((𝐶𝐵) = ∅ → (𝐴𝐶 → ¬ 𝐴𝐵))
54con2d 129 . 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 1523  wcel 2030  cin 3606  c0 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-in 3614  df-nul 3949
This theorem is referenced by: (None)
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