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Theorem linedegen 35115
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) = βˆ…

Proof of Theorem linedegen
Dummy variables 𝑙 𝑛 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 7412 . 2 (𝐴Line𝐴) = (Lineβ€˜βŸ¨π΄, 𝐴⟩)
2 neirr 2950 . . . . . . . . . . 11 Β¬ 𝐴 β‰  𝐴
3 simp3 1139 . . . . . . . . . . 11 ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐴) β†’ 𝐴 β‰  𝐴)
42, 3mto 196 . . . . . . . . . 10 Β¬ (𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐴)
54intnanr 489 . . . . . . . . 9 Β¬ ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]β—‘ Colinear )
65a1i 11 . . . . . . . 8 (𝑛 ∈ β„• β†’ Β¬ ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]β—‘ Colinear ))
76nrex 3075 . . . . . . 7 Β¬ βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]β—‘ Colinear )
87nex 1803 . . . . . 6 Β¬ βˆƒπ‘™βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]β—‘ Colinear )
9 eleq1 2822 . . . . . . . . . . . 12 (π‘₯ = 𝐴 β†’ (π‘₯ ∈ (π”Όβ€˜π‘›) ↔ 𝐴 ∈ (π”Όβ€˜π‘›)))
10 neeq1 3004 . . . . . . . . . . . 12 (π‘₯ = 𝐴 β†’ (π‘₯ β‰  𝑦 ↔ 𝐴 β‰  𝑦))
119, 103anbi13d 1439 . . . . . . . . . . 11 (π‘₯ = 𝐴 β†’ ((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ↔ (𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝑦)))
12 opeq1 4874 . . . . . . . . . . . . 13 (π‘₯ = 𝐴 β†’ ⟨π‘₯, π‘¦βŸ© = ⟨𝐴, π‘¦βŸ©)
1312eceq1d 8742 . . . . . . . . . . . 12 (π‘₯ = 𝐴 β†’ [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear = [⟨𝐴, π‘¦βŸ©]β—‘ Colinear )
1413eqeq2d 2744 . . . . . . . . . . 11 (π‘₯ = 𝐴 β†’ (𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear ↔ 𝑙 = [⟨𝐴, π‘¦βŸ©]β—‘ Colinear ))
1511, 14anbi12d 632 . . . . . . . . . 10 (π‘₯ = 𝐴 β†’ (((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ∧ 𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear ) ↔ ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝑦) ∧ 𝑙 = [⟨𝐴, π‘¦βŸ©]β—‘ Colinear )))
1615rexbidv 3179 . . . . . . . . 9 (π‘₯ = 𝐴 β†’ (βˆƒπ‘› ∈ β„• ((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ∧ 𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear ) ↔ βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝑦) ∧ 𝑙 = [⟨𝐴, π‘¦βŸ©]β—‘ Colinear )))
1716exbidv 1925 . . . . . . . 8 (π‘₯ = 𝐴 β†’ (βˆƒπ‘™βˆƒπ‘› ∈ β„• ((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ∧ 𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear ) ↔ βˆƒπ‘™βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝑦) ∧ 𝑙 = [⟨𝐴, π‘¦βŸ©]β—‘ Colinear )))
18 eleq1 2822 . . . . . . . . . . . 12 (𝑦 = 𝐴 β†’ (𝑦 ∈ (π”Όβ€˜π‘›) ↔ 𝐴 ∈ (π”Όβ€˜π‘›)))
19 neeq2 3005 . . . . . . . . . . . 12 (𝑦 = 𝐴 β†’ (𝐴 β‰  𝑦 ↔ 𝐴 β‰  𝐴))
2018, 193anbi23d 1440 . . . . . . . . . . 11 (𝑦 = 𝐴 β†’ ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝑦) ↔ (𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐴)))
21 opeq2 4875 . . . . . . . . . . . . 13 (𝑦 = 𝐴 β†’ ⟨𝐴, π‘¦βŸ© = ⟨𝐴, 𝐴⟩)
2221eceq1d 8742 . . . . . . . . . . . 12 (𝑦 = 𝐴 β†’ [⟨𝐴, π‘¦βŸ©]β—‘ Colinear = [⟨𝐴, 𝐴⟩]β—‘ Colinear )
2322eqeq2d 2744 . . . . . . . . . . 11 (𝑦 = 𝐴 β†’ (𝑙 = [⟨𝐴, π‘¦βŸ©]β—‘ Colinear ↔ 𝑙 = [⟨𝐴, 𝐴⟩]β—‘ Colinear ))
2420, 23anbi12d 632 . . . . . . . . . 10 (𝑦 = 𝐴 β†’ (((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝑦) ∧ 𝑙 = [⟨𝐴, π‘¦βŸ©]β—‘ Colinear ) ↔ ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]β—‘ Colinear )))
2524rexbidv 3179 . . . . . . . . 9 (𝑦 = 𝐴 β†’ (βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝑦) ∧ 𝑙 = [⟨𝐴, π‘¦βŸ©]β—‘ Colinear ) ↔ βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]β—‘ Colinear )))
2625exbidv 1925 . . . . . . . 8 (𝑦 = 𝐴 β†’ (βˆƒπ‘™βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝑦) ∧ 𝑙 = [⟨𝐴, π‘¦βŸ©]β—‘ Colinear ) ↔ βˆƒπ‘™βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]β—‘ Colinear )))
2717, 26opelopabg 5539 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐴 ∈ V) β†’ (⟨𝐴, 𝐴⟩ ∈ {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘™βˆƒπ‘› ∈ β„• ((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ∧ 𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear )} ↔ βˆƒπ‘™βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]β—‘ Colinear )))
2827anidms 568 . . . . . 6 (𝐴 ∈ V β†’ (⟨𝐴, 𝐴⟩ ∈ {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘™βˆƒπ‘› ∈ β„• ((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ∧ 𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear )} ↔ βˆƒπ‘™βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]β—‘ Colinear )))
298, 28mtbiri 327 . . . . 5 (𝐴 ∈ V β†’ Β¬ ⟨𝐴, 𝐴⟩ ∈ {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘™βˆƒπ‘› ∈ β„• ((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ∧ 𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear )})
30 elopaelxp 5766 . . . . . . 7 (⟨𝐴, 𝐴⟩ ∈ {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘™βˆƒπ‘› ∈ β„• ((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ∧ 𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear )} β†’ ⟨𝐴, 𝐴⟩ ∈ (V Γ— V))
31 opelxp1 5719 . . . . . . 7 (⟨𝐴, 𝐴⟩ ∈ (V Γ— V) β†’ 𝐴 ∈ V)
3230, 31syl 17 . . . . . 6 (⟨𝐴, 𝐴⟩ ∈ {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘™βˆƒπ‘› ∈ β„• ((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ∧ 𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear )} β†’ 𝐴 ∈ V)
3332con3i 154 . . . . 5 (Β¬ 𝐴 ∈ V β†’ Β¬ ⟨𝐴, 𝐴⟩ ∈ {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘™βˆƒπ‘› ∈ β„• ((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ∧ 𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear )})
3429, 33pm2.61i 182 . . . 4 Β¬ ⟨𝐴, 𝐴⟩ ∈ {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘™βˆƒπ‘› ∈ β„• ((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ∧ 𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear )}
35 df-line2 35109 . . . . . . 7 Line = {⟨⟨π‘₯, π‘¦βŸ©, π‘™βŸ© ∣ βˆƒπ‘› ∈ β„• ((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ∧ 𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear )}
3635dmeqi 5905 . . . . . 6 dom Line = dom {⟨⟨π‘₯, π‘¦βŸ©, π‘™βŸ© ∣ βˆƒπ‘› ∈ β„• ((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ∧ 𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear )}
37 dmoprab 7510 . . . . . 6 dom {⟨⟨π‘₯, π‘¦βŸ©, π‘™βŸ© ∣ βˆƒπ‘› ∈ β„• ((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ∧ 𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear )} = {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘™βˆƒπ‘› ∈ β„• ((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ∧ 𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear )}
3836, 37eqtri 2761 . . . . 5 dom Line = {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘™βˆƒπ‘› ∈ β„• ((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ∧ 𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear )}
3938eleq2i 2826 . . . 4 (⟨𝐴, 𝐴⟩ ∈ dom Line ↔ ⟨𝐴, 𝐴⟩ ∈ {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘™βˆƒπ‘› ∈ β„• ((π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›) ∧ π‘₯ β‰  𝑦) ∧ 𝑙 = [⟨π‘₯, π‘¦βŸ©]β—‘ Colinear )})
4034, 39mtbir 323 . . 3 Β¬ ⟨𝐴, 𝐴⟩ ∈ dom Line
41 ndmfv 6927 . . 3 (Β¬ ⟨𝐴, 𝐴⟩ ∈ dom Line β†’ (Lineβ€˜βŸ¨π΄, 𝐴⟩) = βˆ…)
4240, 41ax-mp 5 . 2 (Lineβ€˜βŸ¨π΄, 𝐴⟩) = βˆ…
431, 42eqtri 2761 1 (𝐴Line𝐴) = βˆ…
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2941  βˆƒwrex 3071  Vcvv 3475  βˆ…c0 4323  βŸ¨cop 4635  {copab 5211   Γ— cxp 5675  β—‘ccnv 5676  dom cdm 5677  β€˜cfv 6544  (class class class)co 7409  {coprab 7410  [cec 8701  β„•cn 12212  π”Όcee 28146   Colinear ccolin 35009  Linecline2 35106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fv 6552  df-ov 7412  df-oprab 7413  df-ec 8705  df-line2 35109
This theorem is referenced by: (None)
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