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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isarchi | Structured version Visualization version GIF version |
Description: Express the predicate "𝑊 is Archimedean ". (Contributed by Thierry Arnoux, 30-Jan-2018.) |
Ref | Expression |
---|---|
isarchi.b | ⊢ 𝐵 = (Base‘𝑊) |
isarchi.0 | ⊢ 0 = (0g‘𝑊) |
isarchi.i | ⊢ < = (⋘‘𝑊) |
Ref | Expression |
---|---|
isarchi | ⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveqeq2 6916 | . . 3 ⊢ (𝑤 = 𝑊 → ((⋘‘𝑤) = ∅ ↔ (⋘‘𝑊) = ∅)) | |
2 | df-archi 33169 | . . 3 ⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅} | |
3 | 1, 2 | elab2g 3683 | . 2 ⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ (⋘‘𝑊) = ∅)) |
4 | isarchi.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
5 | 4 | inftmrel 33170 | . . 3 ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵)) |
6 | ss0b 4407 | . . . . 5 ⊢ ((⋘‘𝑊) ⊆ ∅ ↔ (⋘‘𝑊) = ∅) | |
7 | ssrel2 5798 | . . . . 5 ⊢ ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) ⊆ ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅))) | |
8 | 6, 7 | bitr3id 285 | . . . 4 ⊢ ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) = ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅))) |
9 | noel 4344 | . . . . . . . 8 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
10 | 9 | nbn 372 | . . . . . . 7 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
11 | isarchi.i | . . . . . . . . 9 ⊢ < = (⋘‘𝑊) | |
12 | 11 | breqi 5154 | . . . . . . . 8 ⊢ (𝑥 < 𝑦 ↔ 𝑥(⋘‘𝑊)𝑦) |
13 | df-br 5149 | . . . . . . . 8 ⊢ (𝑥(⋘‘𝑊)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊)) | |
14 | 12, 13 | bitri 275 | . . . . . . 7 ⊢ (𝑥 < 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊)) |
15 | 10, 14 | xchnxbir 333 | . . . . . 6 ⊢ (¬ 𝑥 < 𝑦 ↔ (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
16 | 9 | pm2.21i 119 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊)) |
17 | dfbi2 474 | . . . . . . 7 ⊢ ((〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ 〈𝑥, 𝑦〉 ∈ ∅) ↔ ((〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅) ∧ (〈𝑥, 𝑦〉 ∈ ∅ → 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊)))) | |
18 | 16, 17 | mpbiran2 710 | . . . . . 6 ⊢ ((〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ 〈𝑥, 𝑦〉 ∈ ∅) ↔ (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅)) |
19 | 15, 18 | bitri 275 | . . . . 5 ⊢ (¬ 𝑥 < 𝑦 ↔ (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅)) |
20 | 19 | 2ralbii 3126 | . . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅)) |
21 | 8, 20 | bitr4di 289 | . . 3 ⊢ ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) = ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦)) |
22 | 5, 21 | syl 17 | . 2 ⊢ (𝑊 ∈ 𝑉 → ((⋘‘𝑊) = ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦)) |
23 | 3, 22 | bitrd 279 | 1 ⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 ∅c0 4339 〈cop 4637 class class class wbr 5148 × cxp 5687 ‘cfv 6563 Basecbs 17245 0gc0g 17486 ⋘cinftm 33166 Archicarchi 33167 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-inftm 33168 df-archi 33169 |
This theorem is referenced by: xrnarchi 33174 isarchi2 33175 |
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