| 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 6915 | . . 3 ⊢ (𝑤 = 𝑊 → ((⋘‘𝑤) = ∅ ↔ (⋘‘𝑊) = ∅)) | |
| 2 | df-archi 33186 | . . 3 ⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅} | |
| 3 | 1, 2 | elab2g 3680 | . 2 ⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ (⋘‘𝑊) = ∅)) |
| 4 | isarchi.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
| 5 | 4 | inftmrel 33187 | . . 3 ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵)) |
| 6 | ss0b 4401 | . . . . 5 ⊢ ((⋘‘𝑊) ⊆ ∅ ↔ (⋘‘𝑊) = ∅) | |
| 7 | ssrel2 5795 | . . . . 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 4338 | . . . . . . . 8 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
| 10 | 9 | nbn 372 | . . . . . . 7 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
| 11 | isarchi.i | . . . . . . . . 9 ⊢ < = (⋘‘𝑊) | |
| 12 | 11 | breqi 5149 | . . . . . . . 8 ⊢ (𝑥 < 𝑦 ↔ 𝑥(⋘‘𝑊)𝑦) |
| 13 | df-br 5144 | . . . . . . . 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 3128 | . . . 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 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ∅c0 4333 〈cop 4632 class class class wbr 5143 × cxp 5683 ‘cfv 6561 Basecbs 17247 0gc0g 17484 ⋘cinftm 33183 Archicarchi 33184 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-inftm 33185 df-archi 33186 |
| This theorem is referenced by: xrnarchi 33191 isarchi2 33192 |
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