Theorem onsupcl2

Description: The supremum of a set of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025)

	Ref	Expression
Assertion	onsupcl2	$$⊢A\in 𝒫\mathrm{On}\to \bigcup A\in \mathrm{On}$$

Proof

Step	Hyp	Ref	Expression
1		elpwb	$$⊢A\in 𝒫\mathrm{On}\leftrightarrow \left(A\in \mathrm{V}\wedge A\subseteq \mathrm{On}\right)$$
2		ssonuni	$$⊢A\in \mathrm{V}\to \left(A\subseteq \mathrm{On}\to \bigcup A\in \mathrm{On}\right)$$
3	2	imp	$$⊢\left(A\in \mathrm{V}\wedge A\subseteq \mathrm{On}\right)\to \bigcup A\in \mathrm{On}$$
4	1 3	sylbi	$$⊢A\in 𝒫\mathrm{On}\to \bigcup A\in \mathrm{On}$$