Theorem tpid3g

Description: Closed theorem form of tpid3 . (Contributed by Alan Sare, 24-Oct-2011) (Proof shortened by JJ, 30-Apr-2021)

	Ref	Expression
Assertion	tpid3g	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ { 𝐶 , 𝐷 , 𝐴 } )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ 𝐴 = 𝐴
2	1	3mix3i	⊢ ( 𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴 )
3		eltpg	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ { 𝐶 , 𝐷 , 𝐴 } ↔ ( 𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴 ) ) )
4	2 3	mpbiri	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ { 𝐶 , 𝐷 , 𝐴 } )