Metamath Proof Explorer

Theorem ssrankr1

Description: A relationship between an ordinal number less than or equal to a rank, and the cumulative hierarchy of sets R1 . Proposition 9.15(3) of TakeutiZaring p. 79. (Contributed by NM, 8-Oct-2003) (Revised by Mario Carneiro, 17-Nov-2014)

Ref Expression
Hypothesis rankid.1 ⊢ A ∈ V
Assertion ssrankr1 ⊢ B ∈ On → B ⊆ rank ⁡ A ↔ ¬ A ∈ R1 ⁡ B

Proof

Step Hyp Ref Expression
1 rankid.1 ⊢ A ∈ V
2 unir1 ⊢ ⋃ R1 On = V
3 1 2 eleqtrri ⊢ A ∈ ⋃ R1 On
4 r1fnon ⊢ R1 Fn On
5 fndm ⊢ R1 Fn On → dom ⁡ R1 = On
6 4 5 ax-mp ⊢ dom ⁡ R1 = On
7 6 eleq2i ⊢ B ∈ dom ⁡ R1 ↔ B ∈ On
8 7 biimpri ⊢ B ∈ On → B ∈ dom ⁡ R1
9 rankr1clem ⊢ A ∈ ⋃ R1 On ∧ B ∈ dom ⁡ R1 → ¬ A ∈ R1 ⁡ B ↔ B ⊆ rank ⁡ A
10 3 8 9 sylancr ⊢ B ∈ On → ¬ A ∈ R1 ⁡ B ↔ B ⊆ rank ⁡ A
11 10 bicomd ⊢ B ∈ On → B ⊆ rank ⁡ A ↔ ¬ A ∈ R1 ⁡ B