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	$$⊢A\in \mathrm{V}$$
Assertion	eltp	$$⊢A\in \left\{B,C,D\right\}\leftrightarrow \left(A=B\vee A=C\vee A=D\right)$$

Proof

Step	Hyp	Ref	Expression
1		eltp.1	$$⊢A\in \mathrm{V}$$
2		eltpg	$$⊢A\in \mathrm{V}\to \left(A\in \left\{B,C,D\right\}\leftrightarrow \left(A=B\vee A=C\vee A=D\right)\right)$$
3	1 2	ax-mp	$$⊢A\in \left\{B,C,D\right\}\leftrightarrow \left(A=B\vee A=C\vee A=D\right)$$