Theorem rp-intrabeq

Description: Equality theorem for supremum of sets of ordinals. (Contributed by RP, 23-Jan-2025)

	Ref	Expression
Assertion	rp-intrabeq	$$⊢A=B\to \bigcap \left\{x\in \mathrm{On}|\forall y\in Ay\subseteq x\right\}=\bigcap \left\{x\in \mathrm{On}|\forall y\in By\subseteq x\right\}$$

Proof

Step	Hyp	Ref	Expression
1		raleq	$$⊢A=B\to \left(\forall y\in Ay\subseteq x\leftrightarrow \forall y\in By\subseteq x\right)$$
2	1	rabbidv	$$⊢A=B\to \left\{x\in \mathrm{On}|\forall y\in Ay\subseteq x\right\}=\left\{x\in \mathrm{On}|\forall y\in By\subseteq x\right\}$$
3	2	inteqd	$$⊢A=B\to \bigcap \left\{x\in \mathrm{On}|\forall y\in Ay\subseteq x\right\}=\bigcap \left\{x\in \mathrm{On}|\forall y\in By\subseteq x\right\}$$