Theorem tpid2

Description: One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994) (Proof shortened by Andrew Salmon, 29-Jun-2011)

	Ref	Expression
Hypothesis	tpid2.1	⊢ 𝐵 ∈ V
Assertion	tpid2	⊢ 𝐵 ∈ { 𝐴 , 𝐵 , 𝐶 }

Proof

Step	Hyp	Ref	Expression
1		tpid2.1	⊢ 𝐵 ∈ V
2		eqid	⊢ 𝐵 = 𝐵
3	2	3mix2i	⊢ ( 𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶 )
4	1	eltp	⊢ ( 𝐵 ∈ { 𝐴 , 𝐵 , 𝐶 } ↔ ( 𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶 ) )
5	3 4	mpbir	⊢ 𝐵 ∈ { 𝐴 , 𝐵 , 𝐶 }