Theorem elnelne2 2481

Description: Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.)

Assertion

Ref	Expression
elnelne2	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∉ 𝐶) → 𝐴 ≠ 𝐵)

Proof of Theorem elnelne2

Step	Hyp	Ref	Expression
1		df-nel 2472	. 2 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶)
2		nelne2 2467	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵)
3	1, 2	sylan2b 287	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∉ 𝐶) → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∈ wcel 2176   ≠ wne 2376   ∉ wnel 2471

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-cleq 2198  df-clel 2201  df-ne 2377  df-nel 2472

This theorem is referenced by: (None)