Theorem nelne2 2397

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 2200	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
2	1	biimpcd 158	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → 𝐵 ∈ 𝐶))
3	2	necon3bd 2349	. 2 ⊢ (𝐴 ∈ 𝐶 → (¬ 𝐵 ∈ 𝐶 → 𝐴 ≠ 𝐵))
4	3	imp 123	1 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480   ≠ wne 2306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-cleq 2130  df-clel 2133  df-ne 2307

This theorem is referenced by:  nelelne  2398  elnelne2  2411