Users' Mathboxes Mathbox for Rohan Ridenour < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  scottelrankd Structured version   Visualization version   GIF version

Theorem scottelrankd 42939
Description: Property of a Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Hypotheses
Ref Expression
scottelrankd.1 (𝜑𝐵 ∈ Scott 𝐴)
scottelrankd.2 (𝜑𝐶 ∈ Scott 𝐴)
Assertion
Ref Expression
scottelrankd (𝜑 → (rank‘𝐵) ⊆ (rank‘𝐶))

Proof of Theorem scottelrankd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6888 . . 3 (𝑦 = 𝐶 → (rank‘𝑦) = (rank‘𝐶))
21sseq2d 4013 . 2 (𝑦 = 𝐶 → ((rank‘𝐵) ⊆ (rank‘𝑦) ↔ (rank‘𝐵) ⊆ (rank‘𝐶)))
3 scottelrankd.1 . . . . 5 (𝜑𝐵 ∈ Scott 𝐴)
4 df-scott 42928 . . . . 5 Scott 𝐴 = {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
53, 4eleqtrdi 2844 . . . 4 (𝜑𝐵 ∈ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
6 fveq2 6888 . . . . . . 7 (𝑥 = 𝐵 → (rank‘𝑥) = (rank‘𝐵))
76sseq1d 4012 . . . . . 6 (𝑥 = 𝐵 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝐵) ⊆ (rank‘𝑦)))
87ralbidv 3178 . . . . 5 (𝑥 = 𝐵 → (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦𝐴 (rank‘𝐵) ⊆ (rank‘𝑦)))
98elrab 3682 . . . 4 (𝐵 ∈ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ↔ (𝐵𝐴 ∧ ∀𝑦𝐴 (rank‘𝐵) ⊆ (rank‘𝑦)))
105, 9sylib 217 . . 3 (𝜑 → (𝐵𝐴 ∧ ∀𝑦𝐴 (rank‘𝐵) ⊆ (rank‘𝑦)))
1110simprd 497 . 2 (𝜑 → ∀𝑦𝐴 (rank‘𝐵) ⊆ (rank‘𝑦))
12 scottelrankd.2 . . . 4 (𝜑𝐶 ∈ Scott 𝐴)
1312, 4eleqtrdi 2844 . . 3 (𝜑𝐶 ∈ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
14 elrabi 3676 . . 3 (𝐶 ∈ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} → 𝐶𝐴)
1513, 14syl 17 . 2 (𝜑𝐶𝐴)
162, 11, 15rspcdva 3613 1 (𝜑 → (rank‘𝐵) ⊆ (rank‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  {crab 3433  wss 3947  cfv 6540  rankcrnk 9754  Scott cscott 42927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-scott 42928
This theorem is referenced by:  scottrankd  42940
  Copyright terms: Public domain W3C validator