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 6673 | . . 3 ⊢ (𝑤 = 𝑊 → ((⋘‘𝑤) = ∅ ↔ (⋘‘𝑊) = ∅)) | |
2 | df-archi 30803 | . . 3 ⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅} | |
3 | 1, 2 | elab2g 3667 | . 2 ⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ (⋘‘𝑊) = ∅)) |
4 | isarchi.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
5 | 4 | inftmrel 30804 | . . 3 ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵)) |
6 | ss0b 4350 | . . . . 5 ⊢ ((⋘‘𝑊) ⊆ ∅ ↔ (⋘‘𝑊) = ∅) | |
7 | ssrel2 5653 | . . . . 5 ⊢ ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) ⊆ ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅))) | |
8 | 6, 7 | syl5bbr 287 | . . . 4 ⊢ ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) = ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅))) |
9 | noel 4295 | . . . . . . . 8 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
10 | 9 | nbn 375 | . . . . . . 7 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
11 | isarchi.i | . . . . . . . . 9 ⊢ < = (⋘‘𝑊) | |
12 | 11 | breqi 5064 | . . . . . . . 8 ⊢ (𝑥 < 𝑦 ↔ 𝑥(⋘‘𝑊)𝑦) |
13 | df-br 5059 | . . . . . . . 8 ⊢ (𝑥(⋘‘𝑊)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊)) | |
14 | 12, 13 | bitri 277 | . . . . . . 7 ⊢ (𝑥 < 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊)) |
15 | 10, 14 | xchnxbir 335 | . . . . . 6 ⊢ (¬ 𝑥 < 𝑦 ↔ (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
16 | 9 | pm2.21i 119 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊)) |
17 | dfbi2 477 | . . . . . . 7 ⊢ ((〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ 〈𝑥, 𝑦〉 ∈ ∅) ↔ ((〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅) ∧ (〈𝑥, 𝑦〉 ∈ ∅ → 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊)))) | |
18 | 16, 17 | mpbiran2 708 | . . . . . 6 ⊢ ((〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ 〈𝑥, 𝑦〉 ∈ ∅) ↔ (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅)) |
19 | 15, 18 | bitri 277 | . . . . 5 ⊢ (¬ 𝑥 < 𝑦 ↔ (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅)) |
20 | 19 | 2ralbii 3166 | . . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅)) |
21 | 8, 20 | syl6bbr 291 | . . 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 281 | 1 ⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 ∅c0 4290 〈cop 4566 class class class wbr 5058 × cxp 5547 ‘cfv 6349 Basecbs 16477 0gc0g 16707 ⋘cinftm 30800 Archicarchi 30801 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-inftm 30802 df-archi 30803 |
This theorem is referenced by: xrnarchi 30808 isarchi2 30809 |
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