Theorem nelne2 2458

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 2259	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
2	1	biimpcd 159	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 → 𝐵 ∈ 𝐶))
3	2	necon3bd 2410	. 2 ⊢ (𝐴 ∈ 𝐶 → (¬ 𝐵 ∈ 𝐶 → 𝐴 ≠ 𝐵))
4	3	imp 124	1 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167   ≠ wne 2367

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 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192  df-ne 2368

This theorem is referenced by:  nelelne  2459  elnelne2  2472  zgt1rpn0n1  9770