Theorem onsupcl3

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

	Ref	Expression
Assertion	onsupcl3	$$⊢\left(A\subseteq \mathrm{On}\wedge A\in V\right)\to \bigcap \left\{x\in \mathrm{On}|\forall y\in Ay\subseteq x\right\}\in \mathrm{On}$$

Proof

Step	Hyp	Ref	Expression
1		onuniintrab	$$⊢\left(A\subseteq \mathrm{On}\wedge A\in V\right)\to \bigcup A=\bigcap \left\{x\in \mathrm{On}|\forall y\in Ay\subseteq x\right\}$$
2		ssonuni	$$⊢A\in V\to \left(A\subseteq \mathrm{On}\to \bigcup A\in \mathrm{On}\right)$$
3	2	impcom	$$⊢\left(A\subseteq \mathrm{On}\wedge A\in V\right)\to \bigcup A\in \mathrm{On}$$
4	1 3	eqeltrrd	$$⊢\left(A\subseteq \mathrm{On}\wedge A\in V\right)\to \bigcap \left\{x\in \mathrm{On}|\forall y\in Ay\subseteq x\right\}\in \mathrm{On}$$