Theorem scottssr1

Description: Relationship between a Scott's trick set and the cumulative hierarchy. (Contributed by BTernaryTau, 3-Jul-2026)

	Ref	Expression
Assertion	scottssr1	⊢ ( 𝐴 ∈ 𝐵 → Scott 𝐵 ⊆ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rankscottu	⊢ ( 𝐴 ∈ 𝐵 → ( rank ‘ Scott 𝐵 ) ⊆ suc ( rank ‘ 𝐴 ) )
2		rankon	⊢ ( rank ‘ 𝐴 ) ∈ On
3	2	onsuci	⊢ suc ( rank ‘ 𝐴 ) ∈ On
4		scottex	⊢ Scott 𝐵 ∈ V
5	4	rankr1b	⊢ ( suc ( rank ‘ 𝐴 ) ∈ On → ( Scott 𝐵 ⊆ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) ↔ ( rank ‘ Scott 𝐵 ) ⊆ suc ( rank ‘ 𝐴 ) ) )
6	3 5	ax-mp	⊢ ( Scott 𝐵 ⊆ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) ↔ ( rank ‘ Scott 𝐵 ) ⊆ suc ( rank ‘ 𝐴 ) )
7	1 6	sylibr	⊢ ( 𝐴 ∈ 𝐵 → Scott 𝐵 ⊆ ( 𝑅1 ‘ suc ( rank ‘ 𝐴 ) ) )