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Theorem rp-unirabeq 43210
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 3320 . . 3 (𝐴 = 𝐵 → (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦𝐵 𝑥𝑦))
21rabbidv 3440 . 2 (𝐴 = 𝐵 → {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑥𝑦} = {𝑥 ∈ On ∣ ∀𝑦𝐵 𝑥𝑦})
32unieqd 4924 1 (𝐴 = 𝐵 {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑥𝑦} = {𝑥 ∈ On ∣ ∀𝑦𝐵 𝑥𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wral 3058  {crab 3432  wss 3962   cuni 4911  Oncon0 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-ss 3979  df-uni 4912
This theorem is referenced by: (None)
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