Theorem elnelne1 2893

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

Assertion

Ref	Expression
elnelne1	⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∉ 𝐶) → 𝐵 ≠ 𝐶)

Proof of Theorem elnelne1

Step	Hyp	Ref	Expression
1		df-nel 2783	. 2 ⊢ (𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶)
2		nelne1 2878	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐵 ≠ 𝐶)
3	1, 2	sylan2b 491	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∉ 𝐶) → 𝐵 ≠ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606  df-ne 2782  df-nel 2783

This theorem is referenced by: (None)