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Theorem isarchi 31844
Description: Express the predicate "𝑊 is Archimedean ". (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
isarchi.b 𝐵 = (Base‘𝑊)
isarchi.0 0 = (0g𝑊)
isarchi.i < = (⋘‘𝑊)
Assertion
Ref Expression
isarchi (𝑊𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑊,𝑦
Allowed substitution hints:   < (𝑥,𝑦)   𝑉(𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem isarchi
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 fveqeq2 6848 . . 3 (𝑤 = 𝑊 → ((⋘‘𝑤) = ∅ ↔ (⋘‘𝑊) = ∅))
2 df-archi 31841 . . 3 Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
31, 2elab2g 3630 . 2 (𝑊𝑉 → (𝑊 ∈ Archi ↔ (⋘‘𝑊) = ∅))
4 isarchi.b . . . 4 𝐵 = (Base‘𝑊)
54inftmrel 31842 . . 3 (𝑊𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
6 ss0b 4355 . . . . 5 ((⋘‘𝑊) ⊆ ∅ ↔ (⋘‘𝑊) = ∅)
7 ssrel2 5739 . . . . 5 ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) ⊆ ∅ ↔ ∀𝑥𝐵𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅)))
86, 7bitr3id 284 . . . 4 ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) = ∅ ↔ ∀𝑥𝐵𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅)))
9 noel 4288 . . . . . . . 8 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
109nbn 372 . . . . . . 7 (¬ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
11 isarchi.i . . . . . . . . 9 < = (⋘‘𝑊)
1211breqi 5109 . . . . . . . 8 (𝑥 < 𝑦𝑥(⋘‘𝑊)𝑦)
13 df-br 5104 . . . . . . . 8 (𝑥(⋘‘𝑊)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
1412, 13bitri 274 . . . . . . 7 (𝑥 < 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
1510, 14xchnxbir 332 . . . . . 6 𝑥 < 𝑦 ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
169pm2.21i 119 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ∅ → ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
17 dfbi2 475 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅) ∧ (⟨𝑥, 𝑦⟩ ∈ ∅ → ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))))
1816, 17mpbiran2 708 . . . . . 6 ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
1915, 18bitri 274 . . . . 5 𝑥 < 𝑦 ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
20192ralbii 3125 . . . 4 (∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦 ↔ ∀𝑥𝐵𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
218, 20bitr4di 288 . . 3 ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) = ∅ ↔ ∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦))
225, 21syl 17 . 2 (𝑊𝑉 → ((⋘‘𝑊) = ∅ ↔ ∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦))
233, 22bitrd 278 1 (𝑊𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1541  wcel 2106  wral 3062  wss 3908  c0 4280  cop 4590   class class class wbr 5103   × cxp 5629  cfv 6493  Basecbs 17043  0gc0g 17281  cinftm 31838  Archicarchi 31839
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7354  df-inftm 31840  df-archi 31841
This theorem is referenced by:  xrnarchi  31846  isarchi2  31847
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