Theorem elnelne2 3059

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 3049	. 2 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶)
2		nelne2 3041	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵)
3	1, 2	sylan2b 593	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∉ 𝐶) → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2108   ≠ wne 2942   ∉ wnel 3048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-cleq 2730  df-clel 2817  df-ne 2943  df-nel 3049

This theorem is referenced by:  nelrnfvne  6937  eldmrexrnb  6950  absprodnn  16251  frgrncvvdeqlem2  28565  frgrncvvdeqlem3  28566  afv0nbfvbi  44530  uniimaelsetpreimafv  44736  imasetpreimafvbijlemfv1  44743  2zrngnmlid  45395  2zrngnmrid  45396