Theorem linedegen 36878

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 7415	. 2 ⊢ (𝐴Line𝐴) = (Line‘⟨𝐴, 𝐴⟩)
2		neirr 2965	. . . . . . . . . . 11 ⊢ ¬ 𝐴 ≠ 𝐴
3		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) → 𝐴 ≠ 𝐴)
4	2, 3	mto 200	. . . . . . . . . 10 ⊢ ¬ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴)
5	4	intnanr 493	. . . . . . . . 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 3091	. . . . . . 7 ⊢ ¬ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear )
8	7	nex 1833	. . . . . 6 ⊢ ¬ ∃𝑙∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear )
9		eleq1 2849	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
10		neeq1 3018	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝑥 ≠ 𝑦 ↔ 𝐴 ≠ 𝑦))
11	9, 10	3anbi13d 1466	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑦)))
12		opeq1 4833	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
13	12	eceq1d 8742	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → [⟨𝑥, 𝑦⟩]◡ Colinear = [⟨𝐴, 𝑦⟩]◡ Colinear )
14	13	eqeq2d 2772	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear ↔ 𝑙 = [⟨𝐴, 𝑦⟩]◡ Colinear ))
15	11, 14	anbi12d 644	. . . . . . . . . 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 3187	. . . . . . . . 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 1954	. . . . . . . 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 2849	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
19		neeq2 3019	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝐴 ≠ 𝑦 ↔ 𝐴 ≠ 𝐴))
20	18, 19	3anbi23d 1467	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴)))
21		opeq2 4834	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐴⟩)
22	21	eceq1d 8742	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → [⟨𝐴, 𝑦⟩]◡ Colinear = [⟨𝐴, 𝐴⟩]◡ Colinear )
23	22	eqeq2d 2772	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑙 = [⟨𝐴, 𝑦⟩]◡ Colinear ↔ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear ))
24	20, 23	anbi12d 644	. . . . . . . . . 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 3187	. . . . . . . . 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 1954	. . . . . . . 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 5513	. . . . . . 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 577	. . . . . 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 330	. . . . 5 ⊢ (𝐴 ∈ V → ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )})
30		elopaelxp 5741	. . . . . . 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 5693	. . . . . . 7 ⊢ (⟨𝐴, 𝐴⟩ ∈ (V × V) → 𝐴 ∈ V)
32	30, 31	syl 18	. . . . . 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 155	. . . . 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 184	. . . 4 ⊢ ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )}
35		df-line2 36872	. . . . . . 7 ⊢ Line = {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )}
36	35	dmeqi 5886	. . . . . 6 ⊢ dom Line = dom {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )}
37		dmoprab 7515	. . . . . 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 2784	. . . . 5 ⊢ dom Line = {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )}
39	38	eleq2i 2853	. . . 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 326	. . 3 ⊢ ¬ ⟨𝐴, 𝐴⟩ ∈ dom Line
41		ndmfv 6909	. . 3 ⊢ (¬ ⟨𝐴, 𝐴⟩ ∈ dom Line → (Line‘⟨𝐴, 𝐴⟩) = ∅)
42	40, 41	ax-mp 5	. 2 ⊢ (Line‘⟨𝐴, 𝐴⟩) = ∅
43	1, 42	eqtri 2784	1 ⊢ (𝐴Line𝐴) = ∅

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087  Vcvv 3451  ∅c0 4279  ⟨cop 4590  {copab 5167   × cxp 5649  ^◡ccnv 5650  dom cdm 5651  ‘cfv 6531  (class class class)co 7412  {coprab 7413  [cec 8699  ℕcn 12316  𝔼cee 29447   Colinear ccolin 36772  Linecline2 36869

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fv 6539  df-ov 7415  df-oprab 7416  df-ec 8703  df-line2 36872

This theorem is used by: (None)