Theorem elnelne1 3074

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 3064	. 2 ⊢ (𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶)
2		nelne1 3054	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐵 ≠ 𝐶)
3	1, 2	sylan2b 605	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∉ 𝐶) → 𝐵 ≠ 𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   ∈ wcel 2142   ≠ wne 2957   ∉ wnel 3063

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-cleq 2754  df-clel 2837  df-ne 2958  df-nel 3064

This theorem is used by:  sticksstones1  42941