Theorem rankval

Description: Value of the rank function. Definition 9.14 of TakeutiZaring p. 79 (proved as a theorem from our definition). (Contributed by NM, 24-Sep-2003) (Revised by Mario Carneiro, 10-Sep-2013)

	Ref	Expression
Hypothesis	rankval.1	⊢ 𝐴 ∈ V
Assertion	rankval	⊢ ( rank ‘ 𝐴 ) = ∩ { 𝑥 ∈ On ∣ 𝐴 ∈ ( 𝑅1 ‘ suc 𝑥 ) }

Proof

Step	Hyp	Ref	Expression
1		rankval.1	⊢ 𝐴 ∈ V
2		unir1	⊢ ∪ ( 𝑅1 “ On ) = V
3	1 2	eleqtrri	⊢ 𝐴 ∈ ∪ ( 𝑅1 “ On )
4		rankvalb	⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → ( rank ‘ 𝐴 ) = ∩ { 𝑥 ∈ On ∣ 𝐴 ∈ ( 𝑅1 ‘ suc 𝑥 ) } )
5	3 4	ax-mp	⊢ ( rank ‘ 𝐴 ) = ∩ { 𝑥 ∈ On ∣ 𝐴 ∈ ( 𝑅1 ‘ suc 𝑥 ) }