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 ) ) ) |
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 ) ) ) |