Theorem eltp

Description: A member of an unordered triple of classes is one of them. Special case of Exercise 1 of TakeutiZaring p. 17. (Contributed by NM, 8-Apr-1994) (Revised by Mario Carneiro, 11-Feb-2015)

	Ref	Expression
Hypothesis	eltp.1	⊢ 𝐴 ∈ V
Assertion	eltp	⊢ ( 𝐴 ∈ { 𝐵 , 𝐶 , 𝐷 } ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		eltp.1	⊢ 𝐴 ∈ V
2		eltpg	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ { 𝐵 , 𝐶 , 𝐷 } ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ) ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ { 𝐵 , 𝐶 , 𝐷 } ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ) )