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Theorem colinearex 35565
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.
StepHypRef Expression
1 df-colinear 35544 . 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 12222 . . . . 5 ℕ ∈ V
3 fvex 6898 . . . . . . 7 (𝔼‘𝑛) ∈ V
43, 3xpex 7737 . . . . . 6 ((𝔼‘𝑛) × (𝔼‘𝑛)) ∈ V
54, 3xpex 7737 . . . . 5 (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)) ∈ V
62, 5iunex 7954 . . . 4 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)) ∈ V
7 df-oprab 7409 . . . . 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 ∣ ∃𝑏𝑐𝑎(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 5706 . . . . . . . . . . . . . 14 ((𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → ⟨𝑏, 𝑐⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))
983adant1 1127 . . . . . . . . . . . . 13 ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → ⟨𝑏, 𝑐⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))
10 simp1 1133 . . . . . . . . . . . . 13 ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → 𝑎 ∈ (𝔼‘𝑛))
11 opelxpi 5706 . . . . . . . . . . . . 13 ((⟨𝑏, 𝑐⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑎 ∈ (𝔼‘𝑛)) → ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∈ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)))
129, 10, 11syl2anc 583 . . . . . . . . . . . 12 ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∈ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)))
1312adantr 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)))
1413reximi 3078 . . . . . . . . . 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 4994 . . . . . . . . . 10 (⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∈ 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)) ↔ ∃𝑛 ∈ ℕ ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∈ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)))
1614, 15sylibr 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 2815 . . . . . . . . . 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))))
1817biimpar 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)))
1916, 18sylan2 592 . . . . . . . 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)))
2019exlimiv 1925 . . . . . . 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)))
2120exlimivv 1927 . . . . . 6 (∃𝑏𝑐𝑎(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)))
2221abssi 4062 . . . . 5 {𝑥 ∣ ∃𝑏𝑐𝑎(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))
237, 22eqsstri 4011 . . . 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))
246, 23ssexi 5315 . . 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
2524cnvex 7915 . 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
261, 25eqeltri 2823 1 Colinear ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3o 1083  w3a 1084   = wceq 1533  wex 1773  wcel 2098  {cab 2703  wrex 3064  Vcvv 3468  cop 4629   ciun 4990   class class class wbr 5141   × cxp 5667  ccnv 5668  cfv 6537  {coprab 7406  cn 12216  𝔼cee 28654   Btwn cbtwn 28655   Colinear ccolin 35542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-1cn 11170  ax-addcl 11172
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-nn 12217  df-colinear 35544
This theorem is referenced by:  fvline  35649
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