Description: Any set is a subset of the hierarchy of its rank. (Contributed by Mario Carneiro, 3-Jun-2013) (Revised by Mario Carneiro, 17-Nov-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | r1rankidb | ⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → 𝐴 ⊆ ( 𝑅1 ‘ ( rank ‘ 𝐴 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid | ⊢ ( rank ‘ 𝐴 ) ⊆ ( rank ‘ 𝐴 ) | |
2 | rankdmr1 | ⊢ ( rank ‘ 𝐴 ) ∈ dom 𝑅1 | |
3 | rankr1bg | ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝐴 ) ∈ dom 𝑅1 ) → ( 𝐴 ⊆ ( 𝑅1 ‘ ( rank ‘ 𝐴 ) ) ↔ ( rank ‘ 𝐴 ) ⊆ ( rank ‘ 𝐴 ) ) ) | |
4 | 2 3 | mpan2 | ⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → ( 𝐴 ⊆ ( 𝑅1 ‘ ( rank ‘ 𝐴 ) ) ↔ ( rank ‘ 𝐴 ) ⊆ ( rank ‘ 𝐴 ) ) ) |
5 | 1 4 | mpbiri | ⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → 𝐴 ⊆ ( 𝑅1 ‘ ( rank ‘ 𝐴 ) ) ) |