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 6654 | . . 3 ⊢ (𝑤 = 𝑊 → ((⋘‘𝑤) = ∅ ↔ (⋘‘𝑊) = ∅)) | |
2 | df-archi 30858 | . . 3 ⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅} | |
3 | 1, 2 | elab2g 3616 | . 2 ⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ (⋘‘𝑊) = ∅)) |
4 | isarchi.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
5 | 4 | inftmrel 30859 | . . 3 ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵)) |
6 | ss0b 4305 | . . . . 5 ⊢ ((⋘‘𝑊) ⊆ ∅ ↔ (⋘‘𝑊) = ∅) | |
7 | ssrel2 5623 | . . . . 5 ⊢ ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) ⊆ ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅))) | |
8 | 6, 7 | bitr3id 288 | . . . 4 ⊢ ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) = ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅))) |
9 | noel 4247 | . . . . . . . 8 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
10 | 9 | nbn 376 | . . . . . . 7 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
11 | isarchi.i | . . . . . . . . 9 ⊢ < = (⋘‘𝑊) | |
12 | 11 | breqi 5036 | . . . . . . . 8 ⊢ (𝑥 < 𝑦 ↔ 𝑥(⋘‘𝑊)𝑦) |
13 | df-br 5031 | . . . . . . . 8 ⊢ (𝑥(⋘‘𝑊)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊)) | |
14 | 12, 13 | bitri 278 | . . . . . . 7 ⊢ (𝑥 < 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊)) |
15 | 10, 14 | xchnxbir 336 | . . . . . 6 ⊢ (¬ 𝑥 < 𝑦 ↔ (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
16 | 9 | pm2.21i 119 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊)) |
17 | dfbi2 478 | . . . . . . 7 ⊢ ((〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ 〈𝑥, 𝑦〉 ∈ ∅) ↔ ((〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅) ∧ (〈𝑥, 𝑦〉 ∈ ∅ → 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊)))) | |
18 | 16, 17 | mpbiran2 709 | . . . . . 6 ⊢ ((〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ 〈𝑥, 𝑦〉 ∈ ∅) ↔ (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅)) |
19 | 15, 18 | bitri 278 | . . . . 5 ⊢ (¬ 𝑥 < 𝑦 ↔ (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅)) |
20 | 19 | 2ralbii 3134 | . . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅)) |
21 | 8, 20 | syl6bbr 292 | . . 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 282 | 1 ⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 ∅c0 4243 〈cop 4531 class class class wbr 5030 × cxp 5517 ‘cfv 6324 Basecbs 16475 0gc0g 16705 ⋘cinftm 30855 Archicarchi 30856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-inftm 30857 df-archi 30858 |
This theorem is referenced by: xrnarchi 30863 isarchi2 30864 |
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