Theorem onrankid

Description: The rank of an ordinal number is itself. (Contributed by BTernaryTau, 3-Jul-2026)

	Ref	Expression
Assertion	onrankid	$$⊢A\in \mathrm{On}\leftrightarrow \mathrm{rank}\left(A\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		r1fnon	$$⊢R_{1}Fn\mathrm{On}$$
2	1	fndmi	$$⊢\mathrm{dom}{R}_1=\mathrm{On}$$
3	2	eleq2i	$$⊢A\in \mathrm{dom}{R}_1\leftrightarrow A\in \mathrm{On}$$
4		rankonid	$$⊢A\in \mathrm{dom}{R}_1\leftrightarrow \mathrm{rank}\left(A\right)=A$$
5	3 4	bitr3i	$$⊢A\in \mathrm{On}\leftrightarrow \mathrm{rank}\left(A\right)=A$$