Metamath Proof Explorer


Theorem r1rankidb

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 ) ) )

Proof

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 ) ) )