Theorem nelne1 2606

Description: Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)

Assertion

Ref	Expression
nelne1	⊢ ((A ∈ B ∧ ¬ A ∈ C) → B ≠ C)

Proof of Theorem nelne1

Step	Hyp	Ref	Expression
1		eleq2 2414	. . . 4 ⊢ (B = C → (A ∈ B ↔ A ∈ C))
2	1	biimpcd 215	. . 3 ⊢ (A ∈ B → (B = C → A ∈ C))
3	2	necon3bd 2554	. 2 ⊢ (A ∈ B → (¬ A ∈ C → B ≠ C))
4	3	imp 418	1 ⊢ ((A ∈ B ∧ ¬ A ∈ C) → B ≠ C)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2517

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-ne 2519

This theorem is referenced by:  difsnb  3851