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Theorem rp-unirabeq 43234
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 ∣ ∀𝑦𝐴 𝑥𝑦} = {𝑥 ∈ On ∣ ∀𝑦𝐵 𝑥𝑦})
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑥,𝐵   𝑦,𝐵

Proof of Theorem rp-unirabeq
StepHypRef Expression
1 raleq 3323 . . 3 (𝐴 = 𝐵 → (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦𝐵 𝑥𝑦))
21rabbidv 3444 . 2 (𝐴 = 𝐵 → {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑥𝑦} = {𝑥 ∈ On ∣ ∀𝑦𝐵 𝑥𝑦})
32unieqd 4920 1 (𝐴 = 𝐵 {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑥𝑦} = {𝑥 ∈ On ∣ ∀𝑦𝐵 𝑥𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wral 3061  {crab 3436  wss 3951   cuni 4907  Oncon0 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-ss 3968  df-uni 4908
This theorem is referenced by: (None)
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