Theorem onsupintrab

Description: The supremum of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. Definition 2.9 of Schloeder p. 5. (Contributed by RP, 23-Jan-2025)

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

Proof

Step	Hyp	Ref	Expression
1		onsupuni	$$⊢\left(A\subseteq \mathrm{On}\wedge A\in V\right)\to sup\left(A,\mathrm{On},\mathrm{E}\right)=\bigcup A$$
2		onuniintrab	$$⊢\left(A\subseteq \mathrm{On}\wedge A\in V\right)\to \bigcup A=\bigcap \left\{x\in \mathrm{On}|\forall y\in Ay\subseteq x\right\}$$
3	1 2	eqtrd	$$⊢\left(A\subseteq \mathrm{On}\wedge A\in V\right)\to sup\left(A,\mathrm{On},\mathrm{E}\right)=\bigcap \left\{x\in \mathrm{On}|\forall y\in Ay\subseteq x\right\}$$