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

Theorem rp-intrabeq 43209
Description: Equality theorem for supremum of sets of ordinals. (Contributed by RP, 23-Jan-2025.)
Assertion
Ref Expression
rp-intrabeq (𝐴 = 𝐵 {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐵 𝑦𝑥})
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑥,𝐵   𝑦,𝐵

Proof of Theorem rp-intrabeq
StepHypRef Expression
1 raleq 3320 . . 3 (𝐴 = 𝐵 → (∀𝑦𝐴 𝑦𝑥 ↔ ∀𝑦𝐵 𝑦𝑥))
21rabbidv 3440 . 2 (𝐴 = 𝐵 → {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐵 𝑦𝑥})
32inteqd 4955 1 (𝐴 = 𝐵 {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐵 𝑦𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wral 3058  {crab 3432  wss 3962   cint 4950  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-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-ral 3059  df-rex 3068  df-rab 3433  df-int 4951
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator