Theorem onsupex3

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

	Ref	Expression
Assertion	onsupex3	$$⊢\left(A\subseteq \mathrm{On}\wedge A\in V\right)\to \bigcap \left\{x\in \mathrm{On}|\forall y\in Ay\subseteq x\right\}\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		onsupcl3	$$⊢\left(A\subseteq \mathrm{On}\wedge A\in V\right)\to \bigcap \left\{x\in \mathrm{On}|\forall y\in Ay\subseteq x\right\}\in \mathrm{On}$$
2	1	elexd	$$⊢\left(A\subseteq \mathrm{On}\wedge A\in V\right)\to \bigcap \left\{x\in \mathrm{On}|\forall y\in Ay\subseteq x\right\}\in \mathrm{V}$$