Theorem linedegen 35104

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.

Step	Hyp	Ref	Expression
1		df-ov 7409	. 2 ⊢ (𝐴Line𝐴) = (Line‘⟨𝐴, 𝐴⟩)
2		neirr 2950	. . . . . . . . . . 11 ⊢ ¬ 𝐴 ≠ 𝐴
3		simp3 1139	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) → 𝐴 ≠ 𝐴)
4	2, 3	mto 196	. . . . . . . . . 10 ⊢ ¬ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴)
5	4	intnanr 489	. . . . . . . . 9 ⊢ ¬ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear )
6	5	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ¬ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear ))
7	6	nrex 3075	. . . . . . 7 ⊢ ¬ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear )
8	7	nex 1803	. . . . . 6 ⊢ ¬ ∃𝑙∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear )
9		eleq1 2822	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
10		neeq1 3004	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝑥 ≠ 𝑦 ↔ 𝐴 ≠ 𝑦))
11	9, 10	3anbi13d 1439	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑦)))
12		opeq1 4873	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
13	12	eceq1d 8739	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → [⟨𝑥, 𝑦⟩]◡ Colinear = [⟨𝐴, 𝑦⟩]◡ Colinear )
14	13	eqeq2d 2744	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear ↔ 𝑙 = [⟨𝐴, 𝑦⟩]◡ Colinear ))
15	11, 14	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑦) ∧ 𝑙 = [⟨𝐴, 𝑦⟩]◡ Colinear )))
16	15	rexbidv 3179	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑦) ∧ 𝑙 = [⟨𝐴, 𝑦⟩]◡ Colinear )))
17	16	exbidv 1925	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear ) ↔ ∃𝑙∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑦) ∧ 𝑙 = [⟨𝐴, 𝑦⟩]◡ Colinear )))
18		eleq1 2822	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
19		neeq2 3005	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝐴 ≠ 𝑦 ↔ 𝐴 ≠ 𝐴))
20	18, 19	3anbi23d 1440	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴)))
21		opeq2 4874	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐴⟩)
22	21	eceq1d 8739	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → [⟨𝐴, 𝑦⟩]◡ Colinear = [⟨𝐴, 𝐴⟩]◡ Colinear )
23	22	eqeq2d 2744	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑙 = [⟨𝐴, 𝑦⟩]◡ Colinear ↔ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear ))
24	20, 23	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑦) ∧ 𝑙 = [⟨𝐴, 𝑦⟩]◡ Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear )))
25	24	rexbidv 3179	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑦) ∧ 𝑙 = [⟨𝐴, 𝑦⟩]◡ Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear )))
26	25	exbidv 1925	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (∃𝑙∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑦) ∧ 𝑙 = [⟨𝐴, 𝑦⟩]◡ Colinear ) ↔ ∃𝑙∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear )))
27	17, 26	opelopabg 5538	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )} ↔ ∃𝑙∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear )))
28	27	anidms 568	. . . . . 6 ⊢ (𝐴 ∈ V → (⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )} ↔ ∃𝑙∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear )))
29	8, 28	mtbiri 327	. . . . 5 ⊢ (𝐴 ∈ V → ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )})
30		elopaelxp 5764	. . . . . . 7 ⊢ (⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )} → ⟨𝐴, 𝐴⟩ ∈ (V × V))
31		opelxp1 5717	. . . . . . 7 ⊢ (⟨𝐴, 𝐴⟩ ∈ (V × V) → 𝐴 ∈ V)
32	30, 31	syl 17	. . . . . 6 ⊢ (⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )} → 𝐴 ∈ V)
33	32	con3i 154	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )})
34	29, 33	pm2.61i 182	. . . 4 ⊢ ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )}
35		df-line2 35098	. . . . . . 7 ⊢ Line = {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )}
36	35	dmeqi 5903	. . . . . 6 ⊢ dom Line = dom {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )}
37		dmoprab 7507	. . . . . 6 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )}
38	36, 37	eqtri 2761	. . . . 5 ⊢ dom Line = {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )}
39	38	eleq2i 2826	. . . 4 ⊢ (⟨𝐴, 𝐴⟩ ∈ dom Line ↔ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )})
40	34, 39	mtbir 323	. . 3 ⊢ ¬ ⟨𝐴, 𝐴⟩ ∈ dom Line
41		ndmfv 6924	. . 3 ⊢ (¬ ⟨𝐴, 𝐴⟩ ∈ dom Line → (Line‘⟨𝐴, 𝐴⟩) = ∅)
42	40, 41	ax-mp 5	. 2 ⊢ (Line‘⟨𝐴, 𝐴⟩) = ∅
43	1, 42	eqtri 2761	1 ⊢ (𝐴Line𝐴) = ∅

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107   ≠ wne 2941  ∃wrex 3071  Vcvv 3475  ∅c0 4322  ⟨cop 4634  {copab 5210   × cxp 5674  ^◡ccnv 5675  dom cdm 5676  ‘cfv 6541  (class class class)co 7406  {coprab 7407  [cec 8698  ℕcn 12209  𝔼cee 28136   Colinear ccolin 34998  Linecline2 35095

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 5299  ax-nul 5306  ax-pr 5427

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 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fv 6549  df-ov 7409  df-oprab 7410  df-ec 8702  df-line2 35098

This theorem is referenced by: (None)