Theorem df2o3

Description: Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015)

	Ref	Expression
Assertion	df2o3	$$⊢2_{𝑜}=\left\{\varnothing ,1_{𝑜}\right\}$$

Proof

Step	Hyp	Ref	Expression
1		df-2o	$$⊢2_{𝑜}=\mathrm{suc}{1}_{𝑜}$$
2		df-suc	$$⊢\mathrm{suc}{1}_{𝑜}=1_{𝑜}\cup \left\{1_{𝑜}\right\}$$
3		df1o2	$$⊢1_{𝑜}=\left\{\varnothing \right\}$$
4	3	uneq1i	$$⊢1_{𝑜}\cup \left\{1_{𝑜}\right\}=\left\{\varnothing \right\}\cup \left\{1_{𝑜}\right\}$$
5		df-pr	$$⊢\left\{\varnothing ,1_{𝑜}\right\}=\left\{\varnothing \right\}\cup \left\{1_{𝑜}\right\}$$
6	4 5	eqtr4i	$$⊢1_{𝑜}\cup \left\{1_{𝑜}\right\}=\left\{\varnothing ,1_{𝑜}\right\}$$
7	1 2 6	3eqtri	$$⊢2_{𝑜}=\left\{\varnothing ,1_{𝑜}\right\}$$