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Theorem colinearex 32543
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 32522 . 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 11281 . . . . 5 ℕ ∈ V
3 fvex 6388 . . . . . . 7 (𝔼‘𝑛) ∈ V
43, 3xpex 7160 . . . . . 6 ((𝔼‘𝑛) × (𝔼‘𝑛)) ∈ V
54, 3xpex 7160 . . . . 5 (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)) ∈ V
62, 5iunex 7345 . . . 4 𝑛 ∈ ℕ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)) ∈ V
7 df-oprab 6846 . . . . 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 5314 . . . . . . . . . . . . . 14 ((𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → ⟨𝑏, 𝑐⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))
983adant1 1160 . . . . . . . . . . . . 13 ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → ⟨𝑏, 𝑐⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))
10 simp1 1166 . . . . . . . . . . . . 13 ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → 𝑎 ∈ (𝔼‘𝑛))
11 opelxpi 5314 . . . . . . . . . . . . 13 ((⟨𝑏, 𝑐⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑎 ∈ (𝔼‘𝑛)) → ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∈ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)))
129, 10, 11syl2anc 579 . . . . . . . . . . . 12 ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) → ⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∈ (((𝔼‘𝑛) × (𝔼‘𝑛)) × (𝔼‘𝑛)))
1312adantr 472 . . . . . . . . . . 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 3157 . . . . . . . . . 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 4680 . . . . . . . . . 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 225 . . . . . . . . 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 2832 . . . . . . . . . 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 469 . . . . . . . . 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 586 . . . . . . . 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 2025 . . . . . . 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 2027 . . . . . 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 3837 . . . . 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 3795 . . . 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 4964 . . 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 7311 . 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 2840 1 Colinear ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 384  w3o 1106  w3a 1107   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wrex 3056  Vcvv 3350  cop 4340   ciun 4676   class class class wbr 4809   × cxp 5275  ccnv 5276  cfv 6068  {coprab 6843  cn 11274  𝔼cee 26059   Btwn cbtwn 26060   Colinear ccolin 32520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-1cn 10247  ax-addcl 10249
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-nn 11275  df-colinear 32522
This theorem is referenced by:  fvline  32627
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