Theorem rankid

Description: Identity law for the rank function. (Contributed by NM, 3-Oct-2003) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Hypothesis	rankid.1	⊢ 𝐴 ∈ V
Assertion	rankid	⊢ 𝐴 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		rankid.1	⊢ 𝐴 ∈ V
2		unir1	⊢ ∪ ( 𝑅1 “ On ) = V
3	1 2	eleqtrri	⊢ 𝐴 ∈ ∪ ( 𝑅1 “ On )
4		rankidb	⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → 𝐴 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) )
5	3 4	ax-mp	⊢ 𝐴 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) )