Theorem minel 4226

Description: A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.) (Proof shortened by JJ, 14-Jul-2021.)

Assertion

Ref	Expression
minel	⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶)

Proof of Theorem minel

Step	Hyp	Ref	Expression
1		inelcm 4225	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵) → (𝐶 ∩ 𝐵) ≠ ∅)
2	1	expcom 403	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐶 → (𝐶 ∩ 𝐵) ≠ ∅))
3	2	necon2bd 2985	. 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐶 ∩ 𝐵) = ∅ → ¬ 𝐴 ∈ 𝐶))
4	3	imp 396	1 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157   ≠ wne 2969   ∩ cin 3766  ∅c0 4113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-v 3385  df-dif 3770  df-in 3774  df-nul 4114

This theorem is referenced by:  peano5  7321  fnsuppres  7558  domunfican  8473  unwdomg  8729  dfac5  9235  ccatval2  13594  mreexexlem2d  16617  hauspwpwf1  22116