Theorem minel 3350

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	⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶)

Proof of Theorem minel

Step	Hyp	Ref	Expression
1		inelcm 3349	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵) → (𝐶 ∩ 𝐵) ≠ ∅)
2	1	necon2bi 2311	. . . 4 ⊢ ((𝐶 ∩ 𝐵) = ∅ → ¬ (𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵))
3		imnan 660	. . . 4 ⊢ ((𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ 𝐵) ↔ ¬ (𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵))
4	2, 3	sylibr 133	. . 3 ⊢ ((𝐶 ∩ 𝐵) = ∅ → (𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ 𝐵))
5	4	con2d 590	. 2 ⊢ ((𝐶 ∩ 𝐵) = ∅ → (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ 𝐶))
6	5	impcom 124	1 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1290   ∈ wcel 1439   ∩ cin 3001  ∅c0 3289

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-v 2624  df-dif 3004  df-in 3008  df-nul 3290

This theorem is referenced by:  unfidisj  6688  hashunlem  10275