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Metamath Proof Explorer

Theorem elprn1

Description: A member of an unordered pair that is not the "first", must be the "second". (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Assertion elprn1 ⊢ A ∈ B C ∧ A ≠ B → A = C

Proof

Step Hyp Ref Expression
1 elpri ⊢ A ∈ B C → A = B ∨ A = C
2 1 adantr ⊢ A ∈ B C ∧ A ≠ B → A = B ∨ A = C
3 neneq ⊢ A ≠ B → ¬ A = B
4 3 adantl ⊢ A ∈ B C ∧ A ≠ B → ¬ A = B
5 2 4 orcnd ⊢ A ∈ B C ∧ A ≠ B → A = C