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 ∈ ( 𝑅1 ‘ 𝐵 ) ) )

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 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 ) ) )