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Theorem linedegen 32566
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 6873 . 2 (𝐴Line𝐴) = (Line‘⟨𝐴, 𝐴⟩)
2 neirr 2987 . . . . . . . . . . 11 ¬ 𝐴𝐴
3 simp3 1161 . . . . . . . . . . 11 ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) → 𝐴𝐴)
42, 3mto 188 . . . . . . . . . 10 ¬ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴)
54intnanr 477 . . . . . . . . 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 3187 . . . . . . 7 ¬ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear )
87nex 1882 . . . . . 6 ¬ ∃𝑙𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear )
9 eleq1 2873 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑥 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
10 neeq1 3040 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
119, 103anbi13d 1555 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴𝑦)))
12 opeq1 4595 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
1312eceq1d 8014 . . . . . . . . . . . 12 (𝑥 = 𝐴 → [⟨𝑥, 𝑦⟩] Colinear = [⟨𝐴, 𝑦⟩] Colinear )
1413eqeq2d 2816 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝑙 = [⟨𝑥, 𝑦⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝑦⟩] Colinear ))
1511, 14anbi12d 618 . . . . . . . . . 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 3240 . . . . . . . . 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 2012 . . . . . . . 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 2873 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝑦 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
19 neeq2 3041 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝐴𝑦𝐴𝐴))
2018, 193anbi23d 1556 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴)))
21 opeq2 4596 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐴⟩)
2221eceq1d 8014 . . . . . . . . . . . 12 (𝑦 = 𝐴 → [⟨𝐴, 𝑦⟩] Colinear = [⟨𝐴, 𝐴⟩] Colinear )
2322eqeq2d 2816 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑙 = [⟨𝐴, 𝑦⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear ))
2420, 23anbi12d 618 . . . . . . . . . 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 3240 . . . . . . . . 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 2012 . . . . . . . 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 5188 . . . . . . 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 558 . . . . . 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 318 . . . . 5 (𝐴 ∈ V → ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )})
30 elopaelxp 5393 . . . . . . 7 (⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )} → ⟨𝐴, 𝐴⟩ ∈ (V × V))
31 opelxp1 5350 . . . . . . 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 151 . . . . 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 32560 . . . . . . 7 Line = {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
3635dmeqi 5526 . . . . . 6 dom Line = dom {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
37 dmoprab 6967 . . . . . 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 2828 . . . . 5 dom Line = {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
3938eleq2i 2877 . . . 4 (⟨𝐴, 𝐴⟩ ∈ dom Line ↔ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )})
4034, 39mtbir 314 . . 3 ¬ ⟨𝐴, 𝐴⟩ ∈ dom Line
41 ndmfv 6434 . . 3 (¬ ⟨𝐴, 𝐴⟩ ∈ dom Line → (Line‘⟨𝐴, 𝐴⟩) = ∅)
4240, 41ax-mp 5 . 2 (Line‘⟨𝐴, 𝐴⟩) = ∅
431, 42eqtri 2828 1 (𝐴Line𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2156  wne 2978  wrex 3097  Vcvv 3391  c0 4116  cop 4376  {copab 4906   × cxp 5309  ccnv 5310  dom cdm 5311  cfv 6097  (class class class)co 6870  {coprab 6871  [cec 7973  cn 11301  𝔼cee 25978   Colinear ccolin 32460  Linecline2 32557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-xp 5317  df-cnv 5319  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fv 6105  df-ov 6873  df-oprab 6874  df-ec 7977  df-line2 32560
This theorem is referenced by: (None)
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