Theorem elnelne1 3054

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 3044	. 2 ⊢ (𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶)
2		nelne1 3036	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐵 ≠ 𝐶)
3	1, 2	sylan2b 592	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∉ 𝐶) → 𝐵 ≠ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∈ wcel 2098   ≠ wne 2937   ∉ wnel 3043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-cleq 2720  df-clel 2806  df-ne 2938  df-nel 3044

This theorem is referenced by:  sticksstones1  41650