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Theorem linedegen 36144
Description: When Line is applied with the same argument, the result is the empty set. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
linedegen (𝐴Line𝐴) = ∅

Proof of Theorem linedegen
Dummy variables 𝑙 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 7434 . 2 (𝐴Line𝐴) = (Line‘⟨𝐴, 𝐴⟩)
2 neirr 2949 . . . . . . . . . . 11 ¬ 𝐴𝐴
3 simp3 1139 . . . . . . . . . . 11 ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) → 𝐴𝐴)
42, 3mto 197 . . . . . . . . . 10 ¬ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴)
54intnanr 487 . . . . . . . . 9 ¬ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear )
65a1i 11 . . . . . . . 8 (𝑛 ∈ ℕ → ¬ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear ))
76nrex 3074 . . . . . . 7 ¬ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear )
87nex 1800 . . . . . 6 ¬ ∃𝑙𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear )
9 eleq1 2829 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑥 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
10 neeq1 3003 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
119, 103anbi13d 1440 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴𝑦)))
12 opeq1 4873 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
1312eceq1d 8785 . . . . . . . . . . . 12 (𝑥 = 𝐴 → [⟨𝑥, 𝑦⟩] Colinear = [⟨𝐴, 𝑦⟩] Colinear )
1413eqeq2d 2748 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝑙 = [⟨𝑥, 𝑦⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝑦⟩] Colinear ))
1511, 14anbi12d 632 . . . . . . . . . 10 (𝑥 = 𝐴 → (((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴𝑦) ∧ 𝑙 = [⟨𝐴, 𝑦⟩] Colinear )))
1615rexbidv 3179 . . . . . . . . 9 (𝑥 = 𝐴 → (∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴𝑦) ∧ 𝑙 = [⟨𝐴, 𝑦⟩] Colinear )))
1716exbidv 1921 . . . . . . . 8 (𝑥 = 𝐴 → (∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear ) ↔ ∃𝑙𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴𝑦) ∧ 𝑙 = [⟨𝐴, 𝑦⟩] Colinear )))
18 eleq1 2829 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝑦 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
19 neeq2 3004 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝐴𝑦𝐴𝐴))
2018, 193anbi23d 1441 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴)))
21 opeq2 4874 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐴⟩)
2221eceq1d 8785 . . . . . . . . . . . 12 (𝑦 = 𝐴 → [⟨𝐴, 𝑦⟩] Colinear = [⟨𝐴, 𝐴⟩] Colinear )
2322eqeq2d 2748 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑙 = [⟨𝐴, 𝑦⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear ))
2420, 23anbi12d 632 . . . . . . . . . 10 (𝑦 = 𝐴 → (((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴𝑦) ∧ 𝑙 = [⟨𝐴, 𝑦⟩] Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear )))
2524rexbidv 3179 . . . . . . . . 9 (𝑦 = 𝐴 → (∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴𝑦) ∧ 𝑙 = [⟨𝐴, 𝑦⟩] Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear )))
2625exbidv 1921 . . . . . . . 8 (𝑦 = 𝐴 → (∃𝑙𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴𝑦) ∧ 𝑙 = [⟨𝐴, 𝑦⟩] Colinear ) ↔ ∃𝑙𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear )))
2717, 26opelopabg 5543 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )} ↔ ∃𝑙𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear )))
2827anidms 566 . . . . . 6 (𝐴 ∈ V → (⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )} ↔ ∃𝑙𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear )))
298, 28mtbiri 327 . . . . 5 (𝐴 ∈ V → ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )})
30 elopaelxp 5775 . . . . . . 7 (⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )} → ⟨𝐴, 𝐴⟩ ∈ (V × V))
31 opelxp1 5727 . . . . . . 7 (⟨𝐴, 𝐴⟩ ∈ (V × V) → 𝐴 ∈ V)
3230, 31syl 17 . . . . . 6 (⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )} → 𝐴 ∈ V)
3332con3i 154 . . . . 5 𝐴 ∈ V → ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )})
3429, 33pm2.61i 182 . . . 4 ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
35 df-line2 36138 . . . . . . 7 Line = {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
3635dmeqi 5915 . . . . . 6 dom Line = dom {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
37 dmoprab 7536 . . . . . 6 dom {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
3836, 37eqtri 2765 . . . . 5 dom Line = {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
3938eleq2i 2833 . . . 4 (⟨𝐴, 𝐴⟩ ∈ dom Line ↔ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )})
4034, 39mtbir 323 . . 3 ¬ ⟨𝐴, 𝐴⟩ ∈ dom Line
41 ndmfv 6941 . . 3 (¬ ⟨𝐴, 𝐴⟩ ∈ dom Line → (Line‘⟨𝐴, 𝐴⟩) = ∅)
4240, 41ax-mp 5 . 2 (Line‘⟨𝐴, 𝐴⟩) = ∅
431, 42eqtri 2765 1 (𝐴Line𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  wrex 3070  Vcvv 3480  c0 4333  cop 4632  {copab 5205   × cxp 5683  ccnv 5684  dom cdm 5685  cfv 6561  (class class class)co 7431  {coprab 7432  [cec 8743  cn 12266  𝔼cee 28903   Colinear ccolin 36038  Linecline2 36135
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-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569  df-ov 7434  df-oprab 7435  df-ec 8747  df-line2 36138
This theorem is referenced by: (None)
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