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Theorem linedegen 34536
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 7332 . 2 (𝐴Line𝐴) = (Line‘⟨𝐴, 𝐴⟩)
2 neirr 2949 . . . . . . . . . . 11 ¬ 𝐴𝐴
3 simp3 1137 . . . . . . . . . . 11 ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) → 𝐴𝐴)
42, 3mto 196 . . . . . . . . . 10 ¬ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴)
54intnanr 488 . . . . . . . . 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 1801 . . . . . 6 ¬ ∃𝑙𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear )
9 eleq1 2824 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑥 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
10 neeq1 3003 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
119, 103anbi13d 1437 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴𝑦)))
12 opeq1 4816 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
1312eceq1d 8600 . . . . . . . . . . . 12 (𝑥 = 𝐴 → [⟨𝑥, 𝑦⟩] Colinear = [⟨𝐴, 𝑦⟩] Colinear )
1413eqeq2d 2747 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝑙 = [⟨𝑥, 𝑦⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝑦⟩] Colinear ))
1511, 14anbi12d 631 . . . . . . . . . 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 3171 . . . . . . . . 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 1923 . . . . . . . 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 2824 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝑦 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
19 neeq2 3004 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝐴𝑦𝐴𝐴))
2018, 193anbi23d 1438 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴)))
21 opeq2 4817 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐴⟩)
2221eceq1d 8600 . . . . . . . . . . . 12 (𝑦 = 𝐴 → [⟨𝐴, 𝑦⟩] Colinear = [⟨𝐴, 𝐴⟩] Colinear )
2322eqeq2d 2747 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑙 = [⟨𝐴, 𝑦⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear ))
2420, 23anbi12d 631 . . . . . . . . . 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 3171 . . . . . . . . 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 1923 . . . . . . . 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 5476 . . . . . . 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 567 . . . . . 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 326 . . . . 5 (𝐴 ∈ V → ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )})
30 elopaelxp 5701 . . . . . . 7 (⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )} → ⟨𝐴, 𝐴⟩ ∈ (V × V))
31 opelxp1 5655 . . . . . . 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 34530 . . . . . . 7 Line = {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
3635dmeqi 5840 . . . . . 6 dom Line = dom {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
37 dmoprab 7430 . . . . . 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 2764 . . . . 5 dom Line = {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
3938eleq2i 2828 . . . 4 (⟨𝐴, 𝐴⟩ ∈ dom Line ↔ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )})
4034, 39mtbir 322 . . 3 ¬ ⟨𝐴, 𝐴⟩ ∈ dom Line
41 ndmfv 6854 . . 3 (¬ ⟨𝐴, 𝐴⟩ ∈ dom Line → (Line‘⟨𝐴, 𝐴⟩) = ∅)
4240, 41ax-mp 5 . 2 (Line‘⟨𝐴, 𝐴⟩) = ∅
431, 42eqtri 2764 1 (𝐴Line𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396  w3a 1086   = wceq 1540  wex 1780  wcel 2105  wne 2940  wrex 3070  Vcvv 3441  c0 4268  cop 4578  {copab 5151   × cxp 5612  ccnv 5613  dom cdm 5614  cfv 6473  (class class class)co 7329  {coprab 7330  [cec 8559  cn 12066  𝔼cee 27458   Colinear ccolin 34430  Linecline2 34527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fv 6481  df-ov 7332  df-oprab 7333  df-ec 8563  df-line2 34530
This theorem is referenced by: (None)
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