Theorem tpid1

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	tpid1.1	⊢ 𝐴 ∈ V
Assertion	tpid1	⊢ 𝐴 ∈ { 𝐴 , 𝐵 , 𝐶 }

Proof

Step	Hyp	Ref	Expression
1		tpid1.1	⊢ 𝐴 ∈ V
2		eqid	⊢ 𝐴 = 𝐴
3	2	3mix1i	⊢ ( 𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 )
4	1	eltp	⊢ ( 𝐴 ∈ { 𝐴 , 𝐵 , 𝐶 } ↔ ( 𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) )
5	3 4	mpbir	⊢ 𝐴 ∈ { 𝐴 , 𝐵 , 𝐶 }