Theorem nelneq2 2340

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 2302	. . 3 ⊢ (𝐵 = 𝐶 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))
2	1	biimpcd 159	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐵 = 𝐶 → 𝐴 ∈ 𝐶))
3	2	con3dimp 644	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234

This theorem is referenced by:  dtruarb  4323  fzneuz  10486