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Theorem minelOLD 3985
Description: Obsolete proof of minel 3984 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 3983 . . . . 5 ((𝐴𝐶𝐴𝐵) → (𝐶𝐵) ≠ ∅)
21necon2bi 2811 . . . 4 ((𝐶𝐵) = ∅ → ¬ (𝐴𝐶𝐴𝐵))
3 imnan 436 . . . 4 ((𝐴𝐶 → ¬ 𝐴𝐵) ↔ ¬ (𝐴𝐶𝐴𝐵))
42, 3sylibr 222 . . 3 ((𝐶𝐵) = ∅ → (𝐴𝐶 → ¬ 𝐴𝐵))
54con2d 127 . 2 ((𝐶𝐵) = ∅ → (𝐴𝐵 → ¬ 𝐴𝐶))
65impcom 444 1 ((𝐴𝐵 ∧ (𝐶𝐵) = ∅) → ¬ 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1976  cin 3538  c0 3873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-v 3174  df-dif 3542  df-in 3546  df-nul 3874
This theorem is referenced by: (None)
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