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Theorem isarchi 30045
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 6352 . . . 4 (𝑤 = 𝑊 → (⋘‘𝑤) = (⋘‘𝑊))
21eqeq1d 2762 . . 3 (𝑤 = 𝑊 → ((⋘‘𝑤) = ∅ ↔ (⋘‘𝑊) = ∅))
3 df-archi 30042 . . 3 Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
42, 3elab2g 3493 . 2 (𝑊𝑉 → (𝑊 ∈ Archi ↔ (⋘‘𝑊) = ∅))
5 isarchi.b . . . 4 𝐵 = (Base‘𝑊)
65inftmrel 30043 . . 3 (𝑊𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
7 ss0b 4116 . . . . 5 ((⋘‘𝑊) ⊆ ∅ ↔ (⋘‘𝑊) = ∅)
8 ssrel2 5367 . . . . 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 4062 . . . . . . . 8 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
1110nbn 361 . . . . . . 7 (¬ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
12 isarchi.i . . . . . . . . 9 < = (⋘‘𝑊)
1312breqi 4810 . . . . . . . 8 (𝑥 < 𝑦𝑥(⋘‘𝑊)𝑦)
14 df-br 4805 . . . . . . . 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 116 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ∅ → ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
18 dfbi2 663 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅) ∧ (⟨𝑥, 𝑦⟩ ∈ ∅ → ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))))
1917, 18mpbiran2 992 . . . . . 6 ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
2016, 19bitri 264 . . . . 5 𝑥 < 𝑦 ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
21202ralbii 3119 . . . 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 1632  wcel 2139  wral 3050  wss 3715  c0 4058  cop 4327   class class class wbr 4804   × cxp 5264  cfv 6049  Basecbs 16059  0gc0g 16302  cinftm 30039  Archicarchi 30040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-inftm 30041  df-archi 30042
This theorem is referenced by:  xrnarchi  30047  isarchi2  30048
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