Theorem isarchi 33143

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 6870	. . 3 ⊢ (𝑤 = 𝑊 → ((⋘‘𝑤) = ∅ ↔ (⋘‘𝑊) = ∅))
2		df-archi 33140	. . 3 ⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
3	1, 2	elab2g 3650	. 2 ⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ (⋘‘𝑊) = ∅))
4		isarchi.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
5	4	inftmrel 33141	. . 3 ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
6		ss0b 4367	. . . . 5 ⊢ ((⋘‘𝑊) ⊆ ∅ ↔ (⋘‘𝑊) = ∅)
7		ssrel2 5751	. . . . 5 ⊢ ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) ⊆ ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅)))
8	6, 7	bitr3id 285	. . . 4 ⊢ ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) = ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅)))
9		noel 4304	. . . . . . . 8 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
10	9	nbn 372	. . . . . . 7 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
11		isarchi.i	. . . . . . . . 9 ⊢ < = (⋘‘𝑊)
12	11	breqi 5116	. . . . . . . 8 ⊢ (𝑥 < 𝑦 ↔ 𝑥(⋘‘𝑊)𝑦)
13		df-br 5111	. . . . . . . 8 ⊢ (𝑥(⋘‘𝑊)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
14	12, 13	bitri 275	. . . . . . 7 ⊢ (𝑥 < 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
15	10, 14	xchnxbir 333	. . . . . 6 ⊢ (¬ 𝑥 < 𝑦 ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
16	9	pm2.21i 119	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ → ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))
17		dfbi2 474	. . . . . . 7 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅) ∧ (⟨𝑥, 𝑦⟩ ∈ ∅ → ⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊))))
18	16, 17	mpbiran2 710	. . . . . 6 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
19	15, 18	bitri 275	. . . . 5 ⊢ (¬ 𝑥 < 𝑦 ↔ (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
20	19	2ralbii 3109	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ (⋘‘𝑊) → ⟨𝑥, 𝑦⟩ ∈ ∅))
21	8, 20	bitr4di 289	. . 3 ⊢ ((⋘‘𝑊) ⊆ (𝐵 × 𝐵) → ((⋘‘𝑊) = ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦))
22	5, 21	syl 17	. 2 ⊢ (𝑊 ∈ 𝑉 → ((⋘‘𝑊) = ∅ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦))
23	3, 22	bitrd 279	1 ⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ⊆ wss 3917  ∅c0 4299  ⟨cop 4598   class class class wbr 5110   × cxp 5639  ‘cfv 6514  Basecbs 17186  0_gc0g 17409  ⋘cinftm 33137  Archicarchi 33138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-inftm 33139  df-archi 33140

This theorem is referenced by:  xrnarchi  33145  isarchi2  33146