Theorem elnelne2 3050

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 3039	. 2 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶)
2		nelne2 3032	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵)
3	1, 2	sylan2b 600	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∉ 𝐶) → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∈ wcel 2119   ≠ wne 2934   ∉ wnel 3038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2731  df-clel 2814  df-ne 2935  df-nel 3039

This theorem is referenced by:  nelrnfvne  7018  eldmrexrnb  7033  absprodnn  16578  chnrev  18584  frgrncvvdeqlem2  30388  frgrncvvdeqlem3  30389  afv0nbfvbi  47614  uniimaelsetpreimafv  47871  imasetpreimafvbijlemfv1  47878  2zrngnmlid  48746  2zrngnmrid  48747