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Theorem isarchi 29710
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 6178 . . . 4 (𝑤 = 𝑊 → (⋘‘𝑤) = (⋘‘𝑊))
21eqeq1d 2622 . . 3 (𝑤 = 𝑊 → ((⋘‘𝑤) = ∅ ↔ (⋘‘𝑊) = ∅))
3 df-archi 29707 . . 3 Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
42, 3elab2g 3347 . 2 (𝑊𝑉 → (𝑊 ∈ Archi ↔ (⋘‘𝑊) = ∅))
5 isarchi.b . . . 4 𝐵 = (Base‘𝑊)
65inftmrel 29708 . . 3 (𝑊𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
7 ss0b 3964 . . . . 5 ((⋘‘𝑊) ⊆ ∅ ↔ (⋘‘𝑊) = ∅)
8 ssrel2 5200 . . . . 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 3911 . . . . . . . 8 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
1110nbn 362 . . . . . . 7 (¬ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
12 isarchi.i . . . . . . . . 9 < = (⋘‘𝑊)
1312breqi 4650 . . . . . . . 8 (𝑥 < 𝑦𝑥(⋘‘𝑊)𝑦)
14 df-br 4645 . . . . . . . 8 (𝑥(⋘‘𝑊)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
1513, 14bitri 264 . . . . . . 7 (𝑥 < 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
1611, 15xchnxbir 323 . . . . . 6 𝑥 < 𝑦 ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
1710pm2.21i 116 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ∅ → ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
18 dfbi2 659 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅) ∧ (⟨𝑥, 𝑦⟩ ∈ ∅ → ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))))
1917, 18mpbiran2 953 . . . . . 6 ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
2016, 19bitri 264 . . . . 5 𝑥 < 𝑦 ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
21202ralbii 2978 . . . 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 1481  wcel 1988  wral 2909  wss 3567  c0 3907  cop 4174   class class class wbr 4644   × cxp 5102  cfv 5876  Basecbs 15838  0gc0g 16081  cinftm 29704  Archicarchi 29705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-inftm 29706  df-archi 29707
This theorem is referenced by:  xrnarchi  29712  isarchi2  29713
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