Theorem elprn2

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

	Ref	Expression
Assertion	elprn2	⊢ ( ( 𝐴 ∈ { 𝐵 , 𝐶 } ∧ 𝐴 ≠ 𝐶 ) → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		neneq	⊢ ( 𝐴 ≠ 𝐶 → ¬ 𝐴 = 𝐶 )
2	1	adantl	⊢ ( ( 𝐴 ∈ { 𝐵 , 𝐶 } ∧ 𝐴 ≠ 𝐶 ) → ¬ 𝐴 = 𝐶 )
3		elpri	⊢ ( 𝐴 ∈ { 𝐵 , 𝐶 } → ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) )
4	3	adantr	⊢ ( ( 𝐴 ∈ { 𝐵 , 𝐶 } ∧ 𝐴 ≠ 𝐶 ) → ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) )
5		orcom	⊢ ( ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) ↔ ( 𝐴 = 𝐶 ∨ 𝐴 = 𝐵 ) )
6		df-or	⊢ ( ( 𝐴 = 𝐶 ∨ 𝐴 = 𝐵 ) ↔ ( ¬ 𝐴 = 𝐶 → 𝐴 = 𝐵 ) )
7	5 6	bitri	⊢ ( ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) ↔ ( ¬ 𝐴 = 𝐶 → 𝐴 = 𝐵 ) )
8	4 7	sylib	⊢ ( ( 𝐴 ∈ { 𝐵 , 𝐶 } ∧ 𝐴 ≠ 𝐶 ) → ( ¬ 𝐴 = 𝐶 → 𝐴 = 𝐵 ) )
9	2 8	mpd	⊢ ( ( 𝐴 ∈ { 𝐵 , 𝐶 } ∧ 𝐴 ≠ 𝐶 ) → 𝐴 = 𝐵 )