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Metamath Proof Explorer

Theorem nelelne

Description: Two classes are different if they don't belong to the same class. (Contributed by Rodolfo Medina, 17-Oct-2010) (Proof shortened by AV, 10-May-2020)

Ref Expression
Assertion nelelne ⊢ ¬ A ∈ B → C ∈ B → C ≠ A

Proof

Step Hyp Ref Expression
1 nelne2 ⊢ C ∈ B ∧ ¬ A ∈ B → C ≠ A
2 1 expcom ⊢ ¬ A ∈ B → C ∈ B → C ≠ A