Theorem colinearex 32167

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 32146	. 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 11026	. . . . 5 ⊢ ℕ ∈ V
3		fvex 6201	. . . . . . 7 ⊢ (𝔼‘𝑛) ∈ V
4	3, 3	xpex 6962	. . . . . 6 ⊢ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∈ V
5	4, 3	xpex 6962	. . . . 5 ⊢ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)) ∈ V
6	2, 5	iunex 7147	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)) ∈ V
7		df-oprab 6654	. . . . 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 5148	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → ⟨𝑏, 𝑐⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))
9	8	3adant1 1079	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → ⟨𝑏, 𝑐⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))
10		simp1 1061	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → 𝑎 ∈ (𝔼‘𝑛))
11		opelxpi 5148	. . . . . . . . . . . . 13 ⊢ ((⟨𝑏, 𝑐⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑎 ∈ (𝔼‘𝑛)) → ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∈ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)))
12	9, 10, 11	syl2anc 693	. . . . . . . . . . . 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 3011	. . . . . . . . . 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 4524	. . . . . . . . . 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 224	. . . . . . . . 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 2689	. . . . . . . . . 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 502	. . . . . . . . 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 491	. . . . . . . 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 1858	. . . . . . 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 1860	. . . . . 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 3677	. . . . 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 3635	. . . 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 4803	. . 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 7113	. 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 2697	1 ⊢ Colinear ∈ V

Syntax hints:   ∧ wa 384   ∨ w3o 1036   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∃wrex 2913  Vcvv 3200  ⟨cop 4183  ∪ ciun 4520   class class class wbr 4653   × cxp 5112  ^◡ccnv 5113  ‘cfv 5888  {coprab 6651  ℕcn 11020  𝔼cee 25768   Btwn cbtwn 25769   Colinear ccolin 32144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-nn 11021  df-colinear 32146

This theorem is referenced by:  fvline  32251