Theorem minel 3607

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 3606	. . . . 5 ⊢ ((A ∈ C ∧ A ∈ B) → (C ∩ B) ≠ ∅)
2	1	necon2bi 2563	. . . 4 ⊢ ((C ∩ B) = ∅ → ¬ (A ∈ C ∧ A ∈ B))
3		imnan 411	. . . 4 ⊢ ((A ∈ C → ¬ A ∈ B) ↔ ¬ (A ∈ C ∧ A ∈ B))
4	2, 3	sylibr 203	. . 3 ⊢ ((C ∩ B) = ∅ → (A ∈ C → ¬ A ∈ B))
5	4	con2d 107	. 2 ⊢ ((C ∩ B) = ∅ → (A ∈ B → ¬ A ∈ C))
6	5	impcom 419	1 ⊢ ((A ∈ B ∧ (C ∩ B) = ∅) → ¬ A ∈ C)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∩ cin 3209  ∅c0 3551

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is used by: (None)