Theorem tpid2g

Description: Closed theorem form of tpid2 . (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Assertion	tpid2g	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ { 𝐶 , 𝐴 , 𝐷 } )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ 𝐴 = 𝐴
2	1	3mix2i	⊢ ( 𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷 )
3		eltpg	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ { 𝐶 , 𝐴 , 𝐷 } ↔ ( 𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷 ) ) )
4	2 3	mpbiri	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ { 𝐶 , 𝐴 , 𝐷 } )