Theorem tpid2

Description: One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994) (Proof shortened by Andrew Salmon, 29-Jun-2011)

	Ref	Expression
Hypothesis	tpid2.1	$$⊢B\in \mathrm{V}$$
Assertion	tpid2	$$⊢B\in \left\{A,B,C\right\}$$

Proof

Step	Hyp	Ref	Expression
1		tpid2.1	$$⊢B\in \mathrm{V}$$
2		eqid	$$⊢B=B$$
3	2	3mix2i	$$⊢\left(B=A\vee B=B\vee B=C\right)$$
4	1	eltp	$$⊢B\in \left\{A,B,C\right\}\leftrightarrow \left(B=A\vee B=B\vee B=C\right)$$
5	3 4	mpbir	$$⊢B\in \left\{A,B,C\right\}$$