Theorem minel 3556

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 3555	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵) → (𝐶 ∩ 𝐵) ≠ ∅)
2	1	necon2bi 2457	. . . 4 ⊢ ((𝐶 ∩ 𝐵) = ∅ → ¬ (𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵))
3		imnan 696	. . . 4 ⊢ ((𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ 𝐵) ↔ ¬ (𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐵))
4	2, 3	sylibr 134	. . 3 ⊢ ((𝐶 ∩ 𝐵) = ∅ → (𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ 𝐵))
5	4	con2d 629	. 2 ⊢ ((𝐶 ∩ 𝐵) = ∅ → (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ 𝐶))
6	5	impcom 125	1 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202   ∩ cin 3199  ∅c0 3494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-dif 3202  df-in 3206  df-nul 3495

This theorem is referenced by:  unfidisj  7114  hashunlem  11068  ccatval2  11179