Theorem colinearex 36061

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 36040	. 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 12272	. . . . 5 ⊢ ℕ ∈ V
3		fvex 6919	. . . . . . 7 ⊢ (𝔼‘𝑛) ∈ V
4	3, 3	xpex 7773	. . . . . 6 ⊢ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∈ V
5	4, 3	xpex 7773	. . . . 5 ⊢ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)) ∈ V
6	2, 5	iunex 7993	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)) ∈ V
7		df-oprab 7435	. . . . 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 5722	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → ⟨𝑏, 𝑐⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))
9	8	3adant1 1131	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → ⟨𝑏, 𝑐⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))
10		simp1 1137	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → 𝑎 ∈ (𝔼‘𝑛))
11		opelxpi 5722	. . . . . . . . . . . . 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 480	. . . . . . . . . . 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 3084	. . . . . . . . . 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 4995	. . . . . . . . . 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 234	. . . . . . . . 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 2829	. . . . . . . . . 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 477	. . . . . . . . 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 1930	. . . . . . 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 1932	. . . . . 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 4070	. . . . 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 4030	. . . 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 5322	. . 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 7947	. 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 2837	1 ⊢ Colinear ∈ V

Syntax hints:   ∧ wa 395   ∨ w3o 1086   ∧ w3a 1087   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714  ∃wrex 3070  Vcvv 3480  ⟨cop 4632  ∪ ciun 4991   class class class wbr 5143   × cxp 5683  ^◡ccnv 5684  ‘cfv 6561  {coprab 7432  ℕcn 12266  𝔼cee 28903   Btwn cbtwn 28904   Colinear ccolin 36038

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267  df-colinear 36040

This theorem is referenced by:  fvline  36145