Theorem linedegen 36457

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 7395	. 2 ⊢ (𝐴Line𝐴) = (Line‘⟨𝐴, 𝐴⟩)
2		neirr 2965	. . . . . . . . . . 11 ⊢ ¬ 𝐴 ≠ 𝐴
3		simp3 1150	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) → 𝐴 ≠ 𝐴)
4	2, 3	mto 199	. . . . . . . . . 10 ⊢ ¬ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴)
5	4	intnanr 491	. . . . . . . . 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 3089	. . . . . . 7 ⊢ ¬ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear )
8	7	nex 1819	. . . . . 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 1458	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑦)))
12		opeq1 4830	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
13	12	eceq1d 8714	. . . . . . . . . . . 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 641	. . . . . . . . . 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 3185	. . . . . . . . 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 1940	. . . . . . . 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 1459	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴)))
21		opeq2 4831	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐴⟩)
22	21	eceq1d 8714	. . . . . . . . . . . 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 641	. . . . . . . . . 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 3185	. . . . . . . . 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 1940	. . . . . . . 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 5507	. . . . . . 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 574	. . . . . 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 329	. . . . 5 ⊢ (𝐴 ∈ V → ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )})
30		elopaelxp 5735	. . . . . . 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 5687	. . . . . . 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 183	. . . 4 ⊢ ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )}
35		df-line2 36451	. . . . . . 7 ⊢ Line = {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )}
36	35	dmeqi 5878	. . . . . 6 ⊢ dom Line = dom {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )}
37		dmoprab 7495	. . . . . 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 325	. . 3 ⊢ ¬ ⟨𝐴, 𝐴⟩ ∈ dom Line
41		ndmfv 6895	. . 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) = ∅

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559  ∃wex 1798   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3085  Vcvv 3453  ∅c0 4285  ⟨cop 4587  {copab 5161   × cxp 5643  ^◡ccnv 5644  dom cdm 5645  ‘cfv 6517  (class class class)co 7392  {coprab 7393  [cec 8671  ℕcn 12207  𝔼cee 29034   Colinear ccolin 36351  Linecline2 36448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-xp 5651  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fv 6525  df-ov 7395  df-oprab 7396  df-ec 8675  df-line2 36451

This theorem is referenced by: (None)