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Theorem linedegen 31913
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 6610 . 2 (𝐴Line𝐴) = (Line‘⟨𝐴, 𝐴⟩)
2 neirr 2799 . . . . . . . . . . 11 ¬ 𝐴𝐴
3 simp3 1061 . . . . . . . . . . 11 ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) → 𝐴𝐴)
42, 3mto 188 . . . . . . . . . 10 ¬ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴)
54intnanr 960 . . . . . . . . 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 2994 . . . . . . 7 ¬ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear )
87nex 1728 . . . . . 6 ¬ ∃𝑙𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear )
9 eleq1 2686 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑥 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
10 neeq1 2852 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
119, 103anbi13d 1398 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴𝑦)))
12 opeq1 4372 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
1312eceq1d 7731 . . . . . . . . . . . 12 (𝑥 = 𝐴 → [⟨𝑥, 𝑦⟩] Colinear = [⟨𝐴, 𝑦⟩] Colinear )
1413eqeq2d 2631 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝑙 = [⟨𝑥, 𝑦⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝑦⟩] Colinear ))
1511, 14anbi12d 746 . . . . . . . . . 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 3045 . . . . . . . . 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 1847 . . . . . . . 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 2686 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝑦 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
19 neeq2 2853 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝐴𝑦𝐴𝐴))
2018, 193anbi23d 1399 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴)))
21 opeq2 4373 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐴⟩)
2221eceq1d 7731 . . . . . . . . . . . 12 (𝑦 = 𝐴 → [⟨𝐴, 𝑦⟩] Colinear = [⟨𝐴, 𝐴⟩] Colinear )
2322eqeq2d 2631 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑙 = [⟨𝐴, 𝑦⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear ))
2420, 23anbi12d 746 . . . . . . . . . 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 3045 . . . . . . . . 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 1847 . . . . . . . 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 4955 . . . . . . 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 676 . . . . . 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 317 . . . . 5 (𝐴 ∈ V → ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )})
30 elopaelxp 5154 . . . . . . 7 (⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )} → ⟨𝐴, 𝐴⟩ ∈ (V × V))
31 opelxp1 5112 . . . . . . 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 150 . . . . 5 𝐴 ∈ V → ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )})
3429, 33pm2.61i 176 . . . 4 ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
35 df-line2 31907 . . . . . . 7 Line = {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
3635dmeqi 5287 . . . . . 6 dom Line = dom {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
37 dmoprab 6697 . . . . . 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 2643 . . . . 5 dom Line = {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
3938eleq2i 2690 . . . 4 (⟨𝐴, 𝐴⟩ ∈ dom Line ↔ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )})
4034, 39mtbir 313 . . 3 ¬ ⟨𝐴, 𝐴⟩ ∈ dom Line
41 ndmfv 6177 . . 3 (¬ ⟨𝐴, 𝐴⟩ ∈ dom Line → (Line‘⟨𝐴, 𝐴⟩) = ∅)
4240, 41ax-mp 5 . 2 (Line‘⟨𝐴, 𝐴⟩) = ∅
431, 42eqtri 2643 1 (𝐴Line𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wrex 2908  Vcvv 3186  c0 3893  cop 4156  {copab 4674   × cxp 5074  ccnv 5075  dom cdm 5076  cfv 5849  (class class class)co 6607  {coprab 6608  [cec 7688  cn 10967  𝔼cee 25675   Colinear ccolin 31807  Linecline2 31904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-xp 5082  df-cnv 5084  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fv 5857  df-ov 6610  df-oprab 6611  df-ec 7692  df-line2 31907
This theorem is referenced by: (None)
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