Theorem prsnex

Description: The class of preordered sets is a proper class. (Contributed by Zhi Wang, 20-Oct-2025)

	Ref	Expression
Assertion	prsnex	$$⊢\mathrm{Proset}\notin \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		vprc	$$⊢\neg \mathrm{V}\in \mathrm{V}$$
2	1	nelir	$$⊢\mathrm{V}\notin \mathrm{V}$$
3		basresprsfo	$$⊢\left({\mathrm{Base}\upharpoonright }_{\mathrm{Proset}}\right):\mathrm{Proset}\underset{onto}{⟶}\mathrm{V}$$
4	2 3	fonex	$$⊢\mathrm{Proset}\notin \mathrm{V}$$