Theorem minel 4399

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 4398	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵) → (𝐶 ∩ 𝐵) ≠ ∅)
2	1	expcom 414	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐶 → (𝐶 ∩ 𝐵) ≠ ∅))
3	2	necon2bd 2959	. 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐶 ∩ 𝐵) = ∅ → ¬ 𝐴 ∈ 𝐶))
4	3	imp 407	1 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943   ∩ cin 3886  ∅c0 4256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-dif 3890  df-in 3894  df-nul 4257

This theorem is referenced by:  peano5  7740  peano5OLD  7741  fnsuppres  8007  domunfican  9087  unwdomg  9343  dfac5  9884  ccatval2  14283  mreexexlem2d  17354  hauspwpwf1  23138