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 )