Theorem isarchi 33622

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 = (0g‘𝑊)
isarchi.i	⊢ < = (⋘‘𝑊)

Assertion

Ref	Expression
isarchi	⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 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.

Step	Hyp	Ref	Expression
1		fveqeq2 6887	. . 3 ⊢ (𝑤 = 𝑊 → ((⋘‘𝑤) = ∅ ↔ (⋘‘𝑊) = ∅))
2		df-archi 33619	. . 3 ⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
3	1, 2	elab2g 3634	. 2 ⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ (⋘‘𝑊) = ∅))
4		isarchi.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
5	4	inftmrel 33620	. . 3 ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
6		ss0b 4351	. . . . 5 ⊢ ((⋘‘𝑊) ⊆ ∅ ↔ (⋘‘𝑊) = ∅)
7		ssrel2 5765	. . . . 5 ⊢ ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) ⊆ ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅)))
8	6, 7	bitr3id 288	. . . 4 ⊢ ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) = ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅)))
9		noel 4284	. . . . . . . 8 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
10	9	nbn 375	. . . . . . 7 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
11		isarchi.i	. . . . . . . . 9 ⊢ < = (⋘‘𝑊)
12	11	breqi 5109	. . . . . . . 8 ⊢ (𝑥 < 𝑦 ↔ 𝑥(⋘‘𝑊)𝑦)
13		df-br 5104	. . . . . . . 8 ⊢ (𝑥(⋘‘𝑊)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
14	12, 13	bitri 278	. . . . . . 7 ⊢ (𝑥 < 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
15	10, 14	xchnxbir 336	. . . . . 6 ⊢ (¬ 𝑥 < 𝑦 ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
16	9	pm2.21i 120	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ → ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
17		dfbi2 480	. . . . . . 7 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅) ∧ (⟨𝑥, 𝑦⟩ ∈ ∅ → ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))))
18	16, 17	mpbiran2 723	. . . . . 6 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
19	15, 18	bitri 278	. . . . 5 ⊢ (¬ 𝑥 < 𝑦 ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
20	19	2ralbii 3137	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
21	8, 20	bitr4di 292	. . 3 ⊢ ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) = ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦))
22	5, 21	syl 18	. 2 ⊢ (𝑊 ∈ 𝑉 → ((⋘‘𝑊) = ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦))
23	3, 22	bitrd 282	1 ⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ⊆ wss 3899  ∅c0 4279  ⟨cop 4590   class class class wbr 5103   × cxp 5653  ‘cfv 6533  Basecbs 17301  0_gc0g 17524  ⋘cinftm 33616  Archicarchi 33617

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416  df-inftm 33618  df-archi 33619

This theorem is used by:  xrnarchi  33624  isarchi2  33625