Theorem elnelne2 3073

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 3062	. 2 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶)
2		nelne2 3053	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵)
3	1, 2	sylan2b 606	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∉ 𝐶) → 𝐴 ≠ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∈ wcel 2145   ≠ wne 2955   ∉ wnel 3061

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2752  df-clel 2835  df-ne 2956  df-nel 3062

This theorem is used by:  nelrnfvne  7071  eldmrexrnb  7086  absprodnn  16711  chnrev  18718  frgrncvvdeqlem2  30783  frgrncvvdeqlem3  30784  afv0nbfvbi  48042  uniimaelsetpreimafv  48299  imasetpreimafvbijlemfv1  48306  2zrngnmlid  49173  2zrngnmrid  49174