Theorem rp-unirabeq 43579

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

Assertion

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

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

Proof of Theorem rp-unirabeq

Step	Hyp	Ref	Expression
1		raleq 3295	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
2	1	rabbidv 3408	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} = {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦})
3	2	unieqd 4878	1 ⊢ (𝐴 = 𝐵 → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦})

Syntax hints:   → wi 4   = wceq 1542  ∀wral 3052  {crab 3401   ⊆ wss 3903  ∪ cuni 4865  Oncon0 6325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-ss 3920  df-uni 4866

This theorem is referenced by: (None)