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Theorem isarchi 30076
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 fveq2 6332 . . . 4 (𝑤 = 𝑊 → (⋘‘𝑤) = (⋘‘𝑊))
21eqeq1d 2773 . . 3 (𝑤 = 𝑊 → ((⋘‘𝑤) = ∅ ↔ (⋘‘𝑊) = ∅))
3 df-archi 30073 . . 3 Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
42, 3elab2g 3504 . 2 (𝑊𝑉 → (𝑊 ∈ Archi ↔ (⋘‘𝑊) = ∅))
5 isarchi.b . . . 4 𝐵 = (Base‘𝑊)
65inftmrel 30074 . . 3 (𝑊𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
7 ss0b 4117 . . . . 5 ((⋘‘𝑊) ⊆ ∅ ↔ (⋘‘𝑊) = ∅)
8 ssrel2 5350 . . . . 5 ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) ⊆ ∅ ↔ ∀𝑥𝐵𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅)))
97, 8syl5bbr 274 . . . 4 ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) = ∅ ↔ ∀𝑥𝐵𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅)))
10 noel 4067 . . . . . . . 8 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
1110nbn 361 . . . . . . 7 (¬ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
12 isarchi.i . . . . . . . . 9 < = (⋘‘𝑊)
1312breqi 4792 . . . . . . . 8 (𝑥 < 𝑦𝑥(⋘‘𝑊)𝑦)
14 df-br 4787 . . . . . . . 8 (𝑥(⋘‘𝑊)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
1513, 14bitri 264 . . . . . . 7 (𝑥 < 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
1611, 15xchnxbir 322 . . . . . 6 𝑥 < 𝑦 ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
1710pm2.21i 117 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ∅ → ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
18 dfbi2 460 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅) ∧ (⟨𝑥, 𝑦⟩ ∈ ∅ → ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))))
1917, 18mpbiran2 689 . . . . . 6 ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
2016, 19bitri 264 . . . . 5 𝑥 < 𝑦 ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
21202ralbii 3130 . . . 4 (∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦 ↔ ∀𝑥𝐵𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
229, 21syl6bbr 278 . . 3 ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) = ∅ ↔ ∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦))
236, 22syl 17 . 2 (𝑊𝑉 → ((⋘‘𝑊) = ∅ ↔ ∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦))
244, 23bitrd 268 1 (𝑊𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1631  wcel 2145  wral 3061  wss 3723  c0 4063  cop 4322   class class class wbr 4786   × cxp 5247  cfv 6031  Basecbs 16064  0gc0g 16308  cinftm 30070  Archicarchi 30071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-inftm 30072  df-archi 30073
This theorem is referenced by:  xrnarchi  30078  isarchi2  30079
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