Metamath Proof Explorer


Theorem rankr1

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) (Proof shortened by Mario Carneiro, 17-Nov-2014)

Ref Expression
Hypothesis rankid.1 𝐴 ∈ V
Assertion rankr1 ( 𝐵 = ( rank ‘ 𝐴 ) ↔ ( ¬ 𝐴 ∈ ( 𝑅1𝐵 ) ∧ 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 rankid.1 𝐴 ∈ V
2 rankr1g ( 𝐴 ∈ V → ( 𝐵 = ( rank ‘ 𝐴 ) ↔ ( ¬ 𝐴 ∈ ( 𝑅1𝐵 ) ∧ 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) ) ) )
3 1 2 ax-mp ( 𝐵 = ( rank ‘ 𝐴 ) ↔ ( ¬ 𝐴 ∈ ( 𝑅1𝐵 ) ∧ 𝐴 ∈ ( 𝑅1 ‘ suc 𝐵 ) ) )