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Theorem minel 3283
Description: A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)
Assertion
Ref Expression
minel ((𝐴𝐵 ∧ (𝐶𝐵) = ∅) → ¬ 𝐴𝐶)

Proof of Theorem minel
StepHypRef Expression
1 inelcm 3282 . . . . 5 ((𝐴𝐶𝐴𝐵) → (𝐶𝐵) ≠ ∅)
21necon2bi 2260 . . . 4 ((𝐶𝐵) = ∅ → ¬ (𝐴𝐶𝐴𝐵))
3 imnan 624 . . . 4 ((𝐴𝐶 → ¬ 𝐴𝐵) ↔ ¬ (𝐴𝐶𝐴𝐵))
42, 3sylibr 137 . . 3 ((𝐶𝐵) = ∅ → (𝐴𝐶 → ¬ 𝐴𝐵))
54con2d 554 . 2 ((𝐶𝐵) = ∅ → (𝐴𝐵 → ¬ 𝐴𝐶))
65impcom 116 1 ((𝐴𝐵 ∧ (𝐶𝐵) = ∅) → ¬ 𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97   = wceq 1243  wcel 1393  cin 2916  c0 3224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-in 2924  df-nul 3225
This theorem is referenced by: (None)
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