Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rp-unirabeq Structured version   Visualization version   GIF version

Theorem rp-unirabeq 42426
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 3314 . . 3 (𝐴 = 𝐵 → (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦𝐵 𝑥𝑦))
21rabbidv 3432 . 2 (𝐴 = 𝐵 → {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑥𝑦} = {𝑥 ∈ On ∣ ∀𝑦𝐵 𝑥𝑦})
32unieqd 4912 1 (𝐴 = 𝐵 {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑥𝑦} = {𝑥 ∈ On ∣ ∀𝑦𝐵 𝑥𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wral 3053  {crab 3424  wss 3940   cuni 4899  Oncon0 6354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-in 3947  df-ss 3957  df-uni 4900
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator