Theorem nelneq2 2452

Description: A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)

Assertion

Ref	Expression
nelneq2	⊢ ((A ∈ B ∧ ¬ A ∈ C) → ¬ B = C)

Proof of Theorem nelneq2

Step	Hyp	Ref	Expression
1		eleq2 2414	. . 3 ⊢ (B = C → (A ∈ B ↔ A ∈ C))
2	1	biimpcd 215	. 2 ⊢ (A ∈ B → (B = C → A ∈ C))
3	2	con3and 428	1 ⊢ ((A ∈ B ∧ ¬ A ∈ C) → ¬ B = C)

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

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

This theorem is referenced by:  ssnelpss  3614