Theorem eltpi

Description: A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015)

	Ref	Expression
Assertion	eltpi	⊢ ( 𝐴 ∈ { 𝐵 , 𝐶 , 𝐷 } → ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		eltpg	⊢ ( 𝐴 ∈ { 𝐵 , 𝐶 , 𝐷 } → ( 𝐴 ∈ { 𝐵 , 𝐶 , 𝐷 } ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ) ) )
2	1	ibi	⊢ ( 𝐴 ∈ { 𝐵 , 𝐶 , 𝐷 } → ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ) )