Theorem rankr1g

Description: A relationship between the rank function and the cumulative hierarchy of sets function R1 . Proposition 9.15(2) of TakeutiZaring p. 79. (Contributed by NM, 6-Oct-2003) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Assertion	rankr1g	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 = ( rank ‘ 𝐴 ) ↔ ( ¬ 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ∧ 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ V )
2		unir1	⊢ ∪ ( 𝑅1 “ On ) = V
3	1 2	eleqtrrdi	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ ∪ ( 𝑅1 “ On ) )
4		rankr1c	⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → ( 𝐵 = ( rank ‘ 𝐴 ) ↔ ( ¬ 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ∧ 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) ) ) )
5	3 4	syl	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 = ( rank ‘ 𝐴 ) ↔ ( ¬ 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ∧ 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) ) ) )