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Theorem isarchi 30183
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 6384 . . 3 (𝑤 = 𝑊 → ((⋘‘𝑤) = ∅ ↔ (⋘‘𝑊) = ∅))
2 df-archi 30180 . . 3 Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
31, 2elab2g 3508 . 2 (𝑊𝑉 → (𝑊 ∈ Archi ↔ (⋘‘𝑊) = ∅))
4 isarchi.b . . . 4 𝐵 = (Base‘𝑊)
54inftmrel 30181 . . 3 (𝑊𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
6 ss0b 4135 . . . . 5 ((⋘‘𝑊) ⊆ ∅ ↔ (⋘‘𝑊) = ∅)
7 ssrel2 5379 . . . . 5 ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) ⊆ ∅ ↔ ∀𝑥𝐵𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅)))
86, 7syl5bbr 276 . . . 4 ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) = ∅ ↔ ∀𝑥𝐵𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅)))
9 noel 4083 . . . . . . . 8 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
109nbn 363 . . . . . . 7 (¬ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
11 isarchi.i . . . . . . . . 9 < = (⋘‘𝑊)
1211breqi 4815 . . . . . . . 8 (𝑥 < 𝑦𝑥(⋘‘𝑊)𝑦)
13 df-br 4810 . . . . . . . 8 (𝑥(⋘‘𝑊)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
1412, 13bitri 266 . . . . . . 7 (𝑥 < 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
1510, 14xchnxbir 324 . . . . . 6 𝑥 < 𝑦 ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
169pm2.21i 117 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ∅ → ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
17 dfbi2 466 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅) ∧ (⟨𝑥, 𝑦⟩ ∈ ∅ → ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))))
1816, 17mpbiran2 701 . . . . . 6 ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
1915, 18bitri 266 . . . . 5 𝑥 < 𝑦 ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
20192ralbii 3128 . . . 4 (∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦 ↔ ∀𝑥𝐵𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
218, 20syl6bbr 280 . . 3 ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) = ∅ ↔ ∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦))
225, 21syl 17 . 2 (𝑊𝑉 → ((⋘‘𝑊) = ∅ ↔ ∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦))
233, 22bitrd 270 1 (𝑊𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197   = wceq 1652  wcel 2155  wral 3055  wss 3732  c0 4079  cop 4340   class class class wbr 4809   × cxp 5275  cfv 6068  Basecbs 16130  0gc0g 16366  cinftm 30177  Archicarchi 30178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fv 6076  df-ov 6845  df-inftm 30179  df-archi 30180
This theorem is referenced by:  xrnarchi  30185  isarchi2  30186
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