Theorem onsupintrab2

Description: The supremum of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. (Contributed by RP, 23-Jan-2025)

	Ref	Expression
Assertion	onsupintrab2	$$⊢A\in 𝒫\mathrm{On}\to sup\left(A,\mathrm{On},\mathrm{E}\right)=\bigcap \left\{x\in \mathrm{On}|\forall y\in Ay\subseteq x\right\}$$

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		onsupintrab	$$⊢\left(A\subseteq \mathrm{On}\wedge A\in \mathrm{V}\right)\to sup\left(A,\mathrm{On},\mathrm{E}\right)=\bigcap \left\{x\in \mathrm{On}|\forall y\in Ay\subseteq x\right\}$$
3	2	ancoms	$$⊢\left(A\in \mathrm{V}\wedge A\subseteq \mathrm{On}\right)\to sup\left(A,\mathrm{On},\mathrm{E}\right)=\bigcap \left\{x\in \mathrm{On}|\forall y\in Ay\subseteq x\right\}$$
4	1 3	sylbi	$$⊢A\in 𝒫\mathrm{On}\to sup\left(A,\mathrm{On},\mathrm{E}\right)=\bigcap \left\{x\in \mathrm{On}|\forall y\in Ay\subseteq x\right\}$$