Theorem rp-intrabeq

Description: Equality theorem for supremum of sets of ordinals. (Contributed by RP, 23-Jan-2025)

	Ref	Expression
Assertion	rp-intrabeq	⊢ ( 𝐴 = 𝐵 → ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } = ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥 } )

Proof

Step	Hyp	Ref	Expression
1		raleq	⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∀ 𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥 ) )
2	1	rabbidv	⊢ ( 𝐴 = 𝐵 → { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } = { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥 } )
3	2	inteqd	⊢ ( 𝐴 = 𝐵 → ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } = ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥 } )