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		neneq	⊢ ( 𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵 )
2	1	adantl	⊢ ( ( 𝐴 ∈ { 𝐵 , 𝐶 } ∧ 𝐴 ≠ 𝐵 ) → ¬ 𝐴 = 𝐵 )
3		elpri	⊢ ( 𝐴 ∈ { 𝐵 , 𝐶 } → ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) )
4	3	adantr	⊢ ( ( 𝐴 ∈ { 𝐵 , 𝐶 } ∧ 𝐴 ≠ 𝐵 ) → ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) )
5	4	ord	⊢ ( ( 𝐴 ∈ { 𝐵 , 𝐶 } ∧ 𝐴 ≠ 𝐵 ) → ( ¬ 𝐴 = 𝐵 → 𝐴 = 𝐶 ) )
6	2 5	mpd	⊢ ( ( 𝐴 ∈ { 𝐵 , 𝐶 } ∧ 𝐴 ≠ 𝐵 ) → 𝐴 = 𝐶 )