Theorem nelne2 2607

Description: Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)

Assertion

Ref	Expression
nelne2	⊢ ((A ∈ C ∧ ¬ B ∈ C) → A ≠ B)

Proof of Theorem nelne2

Step	Hyp	Ref	Expression
1		eleq1 2413	. . . 4 ⊢ (A = B → (A ∈ C ↔ B ∈ C))
2	1	biimpcd 215	. . 3 ⊢ (A ∈ C → (A = B → B ∈ C))
3	2	necon3bd 2554	. 2 ⊢ (A ∈ C → (¬ B ∈ C → A ≠ B))
4	3	imp 418	1 ⊢ ((A ∈ C ∧ ¬ B ∈ C) → A ≠ B)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2517

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-11 1746  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-ne 2519

This theorem is used by: (None)