Theorem setc1obas

Description: The base of the trivial category. (Contributed by Zhi Wang, 22-Oct-2025)

	Ref	Expression
Hypothesis	funcsetc1o.1	$$⊢\underset{\dot{}}{1}=\mathrm{SetCat}\left(1_{𝑜}\right)$$
Assertion	setc1obas	$$⊢1_{𝑜}={\mathrm{Base}}_{\underset{\dot{}}{1}}$$

Proof

Step	Hyp	Ref	Expression
1		funcsetc1o.1	$$⊢\underset{\dot{}}{1}=\mathrm{SetCat}\left(1_{𝑜}\right)$$
2		1oex	$$⊢1_{𝑜}\in \mathrm{V}$$
3	2	a1i	$$⊢\top \to 1_{𝑜}\in \mathrm{V}$$
4	1 3	setcbas	$$⊢\top \to 1_{𝑜}={\mathrm{Base}}_{\underset{\dot{}}{1}}$$
5	4	mptru	$$⊢1_{𝑜}={\mathrm{Base}}_{\underset{\dot{}}{1}}$$