Theorem rp-intrabeq 43666

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 3294	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥))
2	1	rabbidv 3398	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} = {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥})
3	2	inteqd 4882	1 ⊢ (𝐴 = 𝐵 → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥})

Syntax hints:   → wi 4   = wceq 1547  ∀wral 3053  {crab 3391   ⊆ wss 3883  ∩ cint 4877  Oncon0 6310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-ral 3054  df-rex 3064  df-rab 3392  df-int 4878

This theorem is referenced by: (None)