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 ∈ ( 𝑅1𝐵 ) ) )

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 𝑅1𝐵 ∈ On )
8 7 biimpri ( 𝐵 ∈ On → 𝐵 ∈ dom 𝑅1 )
9 rankr1clem ( ( 𝐴 ( 𝑅1 “ On ) ∧ 𝐵 ∈ dom 𝑅1 ) → ( ¬ 𝐴 ∈ ( 𝑅1𝐵 ) ↔ 𝐵 ⊆ ( rank ‘ 𝐴 ) ) )
10 3 8 9 sylancr ( 𝐵 ∈ On → ( ¬ 𝐴 ∈ ( 𝑅1𝐵 ) ↔ 𝐵 ⊆ ( rank ‘ 𝐴 ) ) )
11 10 bicomd ( 𝐵 ∈ On → ( 𝐵 ⊆ ( rank ‘ 𝐴 ) ↔ ¬ 𝐴 ∈ ( 𝑅1𝐵 ) ) )