Theorem rp-intrabeq 42562

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

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑥,𝐵   𝑦,𝐵

Proof of Theorem rp-intrabeq

Step	Hyp	Ref	Expression
1		raleq 3317	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥))
2	1	rabbidv 3435	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} = {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥})
3	2	inteqd 4949	1 ⊢ (𝐴 = 𝐵 → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥})

Syntax hints:   → wi 4   = wceq 1534  ∀wral 3056  {crab 3427   ⊆ wss 3944  ∩ cint 4944  Oncon0 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-ral 3057  df-rex 3066  df-rab 3428  df-int 4945

This theorem is referenced by: (None)