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 ) ) )