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Theorem scottelrankd 41865
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 6774 . . 3 (𝑦 = 𝐶 → (rank‘𝑦) = (rank‘𝐶))
21sseq2d 3953 . 2 (𝑦 = 𝐶 → ((rank‘𝐵) ⊆ (rank‘𝑦) ↔ (rank‘𝐵) ⊆ (rank‘𝐶)))
3 scottelrankd.1 . . . . 5 (𝜑𝐵 ∈ Scott 𝐴)
4 df-scott 41854 . . . . 5 Scott 𝐴 = {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
53, 4eleqtrdi 2849 . . . 4 (𝜑𝐵 ∈ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
6 fveq2 6774 . . . . . . 7 (𝑥 = 𝐵 → (rank‘𝑥) = (rank‘𝐵))
76sseq1d 3952 . . . . . 6 (𝑥 = 𝐵 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝐵) ⊆ (rank‘𝑦)))
87ralbidv 3112 . . . . 5 (𝑥 = 𝐵 → (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦𝐴 (rank‘𝐵) ⊆ (rank‘𝑦)))
98elrab 3624 . . . 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 496 . 2 (𝜑 → ∀𝑦𝐴 (rank‘𝐵) ⊆ (rank‘𝑦))
12 scottelrankd.2 . . . 4 (𝜑𝐶 ∈ Scott 𝐴)
1312, 4eleqtrdi 2849 . . 3 (𝜑𝐶 ∈ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)})
14 elrabi 3618 . . 3 (𝐶 ∈ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} → 𝐶𝐴)
1513, 14syl 17 . 2 (𝜑𝐶𝐴)
162, 11, 15rspcdva 3562 1 (𝜑 → (rank‘𝐵) ⊆ (rank‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  wral 3064  {crab 3068  wss 3887  cfv 6433  rankcrnk 9521  Scott cscott 41853
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-scott 41854
This theorem is referenced by:  scottrankd  41866
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