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Mirrors > Home > MPE Home > Th. List > Mathboxes > scottelrankd | Structured version Visualization version GIF version |
Description: Property of a Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
Ref | Expression |
---|---|
scottelrankd.1 | ⊢ (𝜑 → 𝐵 ∈ Scott 𝐴) |
scottelrankd.2 | ⊢ (𝜑 → 𝐶 ∈ Scott 𝐴) |
Ref | Expression |
---|---|
scottelrankd | ⊢ (𝜑 → (rank‘𝐵) ⊆ (rank‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6774 | . . 3 ⊢ (𝑦 = 𝐶 → (rank‘𝑦) = (rank‘𝐶)) | |
2 | 1 | sseq2d 3953 | . 2 ⊢ (𝑦 = 𝐶 → ((rank‘𝐵) ⊆ (rank‘𝑦) ↔ (rank‘𝐵) ⊆ (rank‘𝐶))) |
3 | scottelrankd.1 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ Scott 𝐴) | |
4 | df-scott 41854 | . . . . 5 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} | |
5 | 3, 4 | eleqtrdi 2849 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}) |
6 | fveq2 6774 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (rank‘𝑥) = (rank‘𝐵)) | |
7 | 6 | sseq1d 3952 | . . . . . 6 ⊢ (𝑥 = 𝐵 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝐵) ⊆ (rank‘𝑦))) |
8 | 7 | ralbidv 3112 | . . . . 5 ⊢ (𝑥 = 𝐵 → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (rank‘𝐵) ⊆ (rank‘𝑦))) |
9 | 8 | elrab 3624 | . . . 4 ⊢ (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ↔ (𝐵 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (rank‘𝐵) ⊆ (rank‘𝑦))) |
10 | 5, 9 | sylib 217 | . . 3 ⊢ (𝜑 → (𝐵 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (rank‘𝐵) ⊆ (rank‘𝑦))) |
11 | 10 | simprd 496 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (rank‘𝐵) ⊆ (rank‘𝑦)) |
12 | scottelrankd.2 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Scott 𝐴) | |
13 | 12, 4 | eleqtrdi 2849 | . . 3 ⊢ (𝜑 → 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}) |
14 | elrabi 3618 | . . 3 ⊢ (𝐶 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} → 𝐶 ∈ 𝐴) | |
15 | 13, 14 | syl 17 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
16 | 2, 11, 15 | rspcdva 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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