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Theorem linedegen 32588
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 6797 . 2 (𝐴Line𝐴) = (Line‘⟨𝐴, 𝐴⟩)
2 neirr 2952 . . . . . . . . . . 11 ¬ 𝐴𝐴
3 simp3 1132 . . . . . . . . . . 11 ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) → 𝐴𝐴)
42, 3mto 188 . . . . . . . . . 10 ¬ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴)
54intnanr 997 . . . . . . . . 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 3148 . . . . . . 7 ¬ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear )
87nex 1879 . . . . . 6 ¬ ∃𝑙𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear )
9 eleq1 2838 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑥 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
10 neeq1 3005 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
119, 103anbi13d 1549 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴𝑦)))
12 opeq1 4540 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
1312eceq1d 7936 . . . . . . . . . . . 12 (𝑥 = 𝐴 → [⟨𝑥, 𝑦⟩] Colinear = [⟨𝐴, 𝑦⟩] Colinear )
1413eqeq2d 2781 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝑙 = [⟨𝑥, 𝑦⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝑦⟩] Colinear ))
1511, 14anbi12d 610 . . . . . . . . . 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 3200 . . . . . . . . 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 2002 . . . . . . . 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 2838 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝑦 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
19 neeq2 3006 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝐴𝑦𝐴𝐴))
2018, 193anbi23d 1550 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴𝐴)))
21 opeq2 4541 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐴⟩)
2221eceq1d 7936 . . . . . . . . . . . 12 (𝑦 = 𝐴 → [⟨𝐴, 𝑦⟩] Colinear = [⟨𝐴, 𝐴⟩] Colinear )
2322eqeq2d 2781 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑙 = [⟨𝐴, 𝑦⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝐴⟩] Colinear ))
2420, 23anbi12d 610 . . . . . . . . . 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 3200 . . . . . . . . 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 2002 . . . . . . . 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 5127 . . . . . . 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 550 . . . . . 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 316 . . . . 5 (𝐴 ∈ V → ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )})
30 elopaelxp 5332 . . . . . . 7 (⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )} → ⟨𝐴, 𝐴⟩ ∈ (V × V))
31 opelxp1 5291 . . . . . . 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 32582 . . . . . . 7 Line = {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
3635dmeqi 5464 . . . . . 6 dom Line = dom {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
37 dmoprab 6889 . . . . . 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 2793 . . . . 5 dom Line = {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )}
3938eleq2i 2842 . . . 4 (⟨𝐴, 𝐴⟩ ∈ dom Line ↔ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩] Colinear )})
4034, 39mtbir 312 . . 3 ¬ ⟨𝐴, 𝐴⟩ ∈ dom Line
41 ndmfv 6360 . . 3 (¬ ⟨𝐴, 𝐴⟩ ∈ dom Line → (Line‘⟨𝐴, 𝐴⟩) = ∅)
4240, 41ax-mp 5 . 2 (Line‘⟨𝐴, 𝐴⟩) = ∅
431, 42eqtri 2793 1 (𝐴Line𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  wne 2943  wrex 3062  Vcvv 3351  c0 4064  cop 4323  {copab 4847   × cxp 5248  ccnv 5249  dom cdm 5250  cfv 6032  (class class class)co 6794  {coprab 6795  [cec 7895  cn 11223  𝔼cee 25990   Colinear ccolin 32482  Linecline2 32579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fv 6040  df-ov 6797  df-oprab 6798  df-ec 7899  df-line2 32582
This theorem is referenced by: (None)
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