Metamath Proof Explorer

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