Theorem colinearex 34577

Description: The colinear predicate exists. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
colinearex	⊢ Colinear ∈ V

Proof of Theorem colinearex

Dummy variables 𝑎 𝑏 𝑐 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-colinear 34556	. 2 ⊢ Colinear = ◡{⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))}
2		nnex 12117	. . . . 5 ⊢ ℕ ∈ V
3		fvex 6852	. . . . . . 7 ⊢ (𝔼‘𝑛) ∈ V
4	3, 3	xpex 7679	. . . . . 6 ⊢ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∈ V
5	4, 3	xpex 7679	. . . . 5 ⊢ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)) ∈ V
6	2, 5	iunex 7893	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)) ∈ V
7		df-oprab 7355	. . . . 5 ⊢ {⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))} = {𝑥 ∣ ∃𝑏∃𝑐∃𝑎(𝑥 = ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∧ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩)))}
8		opelxpi 5668	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → ⟨𝑏, 𝑐⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))
9	8	3adant1 1130	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → ⟨𝑏, 𝑐⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))
10		simp1 1136	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → 𝑎 ∈ (𝔼‘𝑛))
11		opelxpi 5668	. . . . . . . . . . . . 13 ⊢ ((⟨𝑏, 𝑐⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑎 ∈ (𝔼‘𝑛)) → ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∈ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)))
12	9, 10, 11	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∈ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)))
13	12	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩)) → ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∈ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)))
14	13	reximi 3085	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩)) → ∃𝑛 ∈ ℕ ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∈ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)))
15		eliun 4956	. . . . . . . . . 10 ⊢ (⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∈ ∪ 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)) ↔ ∃𝑛 ∈ ℕ ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∈ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)))
16	14, 15	sylibr 233	. . . . . . . . 9 ⊢ (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩)) → ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∈ ∪ 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)))
17		eleq1 2825	. . . . . . . . . 10 ⊢ (𝑥 = ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ → (𝑥 ∈ ∪ 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)) ↔ ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∈ ∪ 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛))))
18	17	biimpar 478	. . . . . . . . 9 ⊢ ((𝑥 = ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∧ ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∈ ∪ 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛))) → 𝑥 ∈ ∪ 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)))
19	16, 18	sylan2 593	. . . . . . . 8 ⊢ ((𝑥 = ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∧ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))) → 𝑥 ∈ ∪ 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)))
20	19	exlimiv 1933	. . . . . . 7 ⊢ (∃𝑎(𝑥 = ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∧ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))) → 𝑥 ∈ ∪ 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)))
21	20	exlimivv 1935	. . . . . 6 ⊢ (∃𝑏∃𝑐∃𝑎(𝑥 = ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∧ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))) → 𝑥 ∈ ∪ 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)))
22	21	abssi 4025	. . . . 5 ⊢ {𝑥 ∣ ∃𝑏∃𝑐∃𝑎(𝑥 = ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∧ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩)))} ⊆ ∪ 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛))
23	7, 22	eqsstri 3976	. . . 4 ⊢ {⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))} ⊆ ∪ 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛))
24	6, 23	ssexi 5277	. . 3 ⊢ {⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))} ∈ V
25	24	cnvex 7854	. 2 ⊢ ◡{⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))} ∈ V
26	1, 25	eqeltri 2834	1 ⊢ Colinear ∈ V

Syntax hints:   ∧ wa 396   ∨ w3o 1086   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2713  ∃wrex 3071  Vcvv 3443  ⟨cop 4590  ∪ ciun 4952   class class class wbr 5103   × cxp 5629  ^◡ccnv 5630  ‘cfv 6493  {coprab 7352  ℕcn 12111  𝔼cee 27682   Btwn cbtwn 27683   Colinear ccolin 34554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-1cn 11067  ax-addcl 11069

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-om 7795  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-nn 12112  df-colinear 34556

This theorem is referenced by:  fvline  34661