Theorem nelne1 2426

Description: Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)

Assertion

Ref	Expression
nelne1	⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐵 ≠ 𝐶)

Proof of Theorem nelne1

Step	Hyp	Ref	Expression
1		eleq2 2230	. . . 4 ⊢ (𝐵 = 𝐶 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))
2	1	biimpcd 158	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐵 = 𝐶 → 𝐴 ∈ 𝐶))
3	2	necon3bd 2379	. 2 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ 𝐶 → 𝐵 ≠ 𝐶))
4	3	imp 123	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐵 ≠ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136   ≠ wne 2336

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 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-clel 2161  df-ne 2337

This theorem is referenced by:  elnelne1  2440  difsnb  3716