Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 6726 | . . 3 ⊢ (𝑤 = 𝑊 → ((⋘‘𝑤) = ∅ ↔ (⋘‘𝑊) = ∅)) | |
2 | df-archi 31152 | . . 3 ⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅} | |
3 | 1, 2 | elab2g 3589 | . 2 ⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ (⋘‘𝑊) = ∅)) |
4 | isarchi.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
5 | 4 | inftmrel 31153 | . . 3 ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵)) |
6 | ss0b 4312 | . . . . 5 ⊢ ((⋘‘𝑊) ⊆ ∅ ↔ (⋘‘𝑊) = ∅) | |
7 | ssrel2 5656 | . . . . 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 4245 | . . . . . . . 8 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
10 | 9 | nbn 376 | . . . . . . 7 ⊢ (¬ 〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) ↔ 〈𝑥, 𝑦〉 ∈ ∅)) |
11 | isarchi.i | . . . . . . . . 9 ⊢ < = (⋘‘𝑊) | |
12 | 11 | breqi 5059 | . . . . . . . 8 ⊢ (𝑥 < 𝑦 ↔ 𝑥(⋘‘𝑊)𝑦) |
13 | df-br 5054 | . . . . . . . 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 710 | . . . . . 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 3089 | . . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (〈𝑥, 𝑦〉 ∈ (⋘‘𝑊) → 〈𝑥, 𝑦〉 ∈ ∅)) |
21 | 8, 20 | bitr4di 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 1543 ∈ wcel 2110 ∀wral 3061 ⊆ wss 3866 ∅c0 4237 〈cop 4547 class class class wbr 5053 × cxp 5549 ‘cfv 6380 Basecbs 16760 0gc0g 16944 ⋘cinftm 31149 Archicarchi 31150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-ov 7216 df-inftm 31151 df-archi 31152 |
This theorem is referenced by: xrnarchi 31157 isarchi2 31158 |
Copyright terms: Public domain | W3C validator |