Theorem tpid1

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	tpid1.1	$$⊢A\in \mathrm{V}$$
Assertion	tpid1	$$⊢A\in \left\{A,B,C\right\}$$

Proof

Step	Hyp	Ref	Expression
1		tpid1.1	$$⊢A\in \mathrm{V}$$
2		eqid	$$⊢A=A$$
3	2	3mix1i	$$⊢\left(A=A\vee A=B\vee A=C\right)$$
4	1	eltp	$$⊢A\in \left\{A,B,C\right\}\leftrightarrow \left(A=A\vee A=B\vee A=C\right)$$
5	3 4	mpbir	$$⊢A\in \left\{A,B,C\right\}$$