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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 6663 | . . 3 ⊢ (𝑦 = 𝐶 → (rank‘𝑦) = (rank‘𝐶)) | |
2 | 1 | sseq2d 3992 | . 2 ⊢ (𝑦 = 𝐶 → ((rank‘𝐵) ⊆ (rank‘𝑦) ↔ (rank‘𝐵) ⊆ (rank‘𝐶))) |
3 | scottelrankd.1 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ Scott 𝐴) | |
4 | df-scott 40646 | . . . . 5 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} | |
5 | 3, 4 | eleqtrdi 2922 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}) |
6 | fveq2 6663 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (rank‘𝑥) = (rank‘𝐵)) | |
7 | 6 | sseq1d 3991 | . . . . . 6 ⊢ (𝑥 = 𝐵 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝐵) ⊆ (rank‘𝑦))) |
8 | 7 | ralbidv 3196 | . . . . 5 ⊢ (𝑥 = 𝐵 → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (rank‘𝐵) ⊆ (rank‘𝑦))) |
9 | 8 | elrab 3676 | . . . 4 ⊢ (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ↔ (𝐵 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (rank‘𝐵) ⊆ (rank‘𝑦))) |
10 | 5, 9 | sylib 220 | . . 3 ⊢ (𝜑 → (𝐵 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (rank‘𝐵) ⊆ (rank‘𝑦))) |
11 | 10 | simprd 498 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (rank‘𝐵) ⊆ (rank‘𝑦)) |
12 | scottelrankd.2 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Scott 𝐴) | |
13 | 12, 4 | eleqtrdi 2922 | . . 3 ⊢ (𝜑 → 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}) |
14 | elrabi 3671 | . . 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 3622 | 1 ⊢ (𝜑 → (rank‘𝐵) ⊆ (rank‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 {crab 3141 ⊆ wss 3929 ‘cfv 6348 rankcrnk 9185 Scott cscott 40645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-iota 6307 df-fv 6356 df-scott 40646 |
This theorem is referenced by: scottrankd 40658 |
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