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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 ∈ R1 ⁡ B ∧ A ∈ R1 ⁡ suc ⁡ B

Proof

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