Theorem nelneq 2451

Description: A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)

Assertion

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

Proof of Theorem nelneq

Step	Hyp	Ref	Expression
1		eleq1 2413	. . 3 ⊢ (A = B → (A ∈ C ↔ B ∈ C))
2	1	biimpcd 215	. 2 ⊢ (A ∈ C → (A = B → B ∈ C))
3	2	con3and 428	1 ⊢ ((A ∈ C ∧ ¬ B ∈ C) → ¬ A = B)

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: (None)