Theorem onrankid

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

	Ref	Expression
Assertion	onrankid	⊢ ( 𝐴 ∈ On ↔ ( rank ‘ 𝐴 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		r1fnon	⊢ 𝑅1 Fn On
2	1	fndmi	⊢ dom 𝑅1 = On
3	2	eleq2i	⊢ ( 𝐴 ∈ dom 𝑅1 ↔ 𝐴 ∈ On )
4		rankonid	⊢ ( 𝐴 ∈ dom 𝑅1 ↔ ( rank ‘ 𝐴 ) = 𝐴 )
5	3 4	bitr3i	⊢ ( 𝐴 ∈ On ↔ ( rank ‘ 𝐴 ) = 𝐴 )