Theorem rp-unirabeq

Description: Equality theorem for infimum of non-empty classes of ordinals. (Contributed by RP, 23-Jan-2025)

	Ref	Expression
Assertion	rp-unirabeq	$$⊢A=B\to \bigcup \left\{x\in \mathrm{On}|\forall y\in Ax\subseteq y\right\}=\bigcup \left\{x\in \mathrm{On}|\forall y\in Bx\subseteq y\right\}$$

Proof

Step	Hyp	Ref	Expression
1		raleq	$$⊢A=B\to \left(\forall y\in Ax\subseteq y\leftrightarrow \forall y\in Bx\subseteq y\right)$$
2	1	rabbidv	$$⊢A=B\to \left\{x\in \mathrm{On}|\forall y\in Ax\subseteq y\right\}=\left\{x\in \mathrm{On}|\forall y\in Bx\subseteq y\right\}$$
3	2	unieqd	$$⊢A=B\to \bigcup \left\{x\in \mathrm{On}|\forall y\in Ax\subseteq y\right\}=\bigcup \left\{x\in \mathrm{On}|\forall y\in Bx\subseteq y\right\}$$