Theorem rp-unirabeq

Description: Equality theorem for infimum of non-empty classes of ordinals. (Contributed by RP, 23-Jan-2025)

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

Proof

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