Theorem linedegen 32566

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 6873	. 2 ⊢ (𝐴Line𝐴) = (Line‘⟨𝐴, 𝐴⟩)
2		neirr 2987	. . . . . . . . . . 11 ⊢ ¬ 𝐴 ≠ 𝐴
3		simp3 1161	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) → 𝐴 ≠ 𝐴)
4	2, 3	mto 188	. . . . . . . . . 10 ⊢ ¬ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴)
5	4	intnanr 477	. . . . . . . . 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 3187	. . . . . . 7 ⊢ ¬ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear )
8	7	nex 1882	. . . . . 6 ⊢ ¬ ∃𝑙∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴) ∧ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear )
9		eleq1 2873	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
10		neeq1 3040	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝑥 ≠ 𝑦 ↔ 𝐴 ≠ 𝑦))
11	9, 10	3anbi13d 1555	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑦)))
12		opeq1 4595	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
13	12	eceq1d 8014	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → [⟨𝑥, 𝑦⟩]◡ Colinear = [⟨𝐴, 𝑦⟩]◡ Colinear )
14	13	eqeq2d 2816	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear ↔ 𝑙 = [⟨𝐴, 𝑦⟩]◡ Colinear ))
15	11, 14	anbi12d 618	. . . . . . . . . 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 3240	. . . . . . . . 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 2012	. . . . . . . 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 2873	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
19		neeq2 3041	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝐴 ≠ 𝑦 ↔ 𝐴 ≠ 𝐴))
20	18, 19	3anbi23d 1556	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑦) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐴)))
21		opeq2 4596	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐴⟩)
22	21	eceq1d 8014	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → [⟨𝐴, 𝑦⟩]◡ Colinear = [⟨𝐴, 𝐴⟩]◡ Colinear )
23	22	eqeq2d 2816	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑙 = [⟨𝐴, 𝑦⟩]◡ Colinear ↔ 𝑙 = [⟨𝐴, 𝐴⟩]◡ Colinear ))
24	20, 23	anbi12d 618	. . . . . . . . . 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 3240	. . . . . . . . 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 2012	. . . . . . . 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 5188	. . . . . . 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 558	. . . . . 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 318	. . . . 5 ⊢ (𝐴 ∈ V → ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )})
30		elopaelxp 5393	. . . . . . 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 5350	. . . . . . 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 151	. . . . 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 176	. . . 4 ⊢ ¬ ⟨𝐴, 𝐴⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )}
35		df-line2 32560	. . . . . . 7 ⊢ Line = {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )}
36	35	dmeqi 5526	. . . . . 6 ⊢ dom Line = dom {⟨⟨𝑥, 𝑦⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )}
37		dmoprab 6967	. . . . . 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 2828	. . . . 5 ⊢ dom Line = {⟨𝑥, 𝑦⟩ ∣ ∃𝑙∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛) ∧ 𝑥 ≠ 𝑦) ∧ 𝑙 = [⟨𝑥, 𝑦⟩]◡ Colinear )}
39	38	eleq2i 2877	. . . 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 314	. . 3 ⊢ ¬ ⟨𝐴, 𝐴⟩ ∈ dom Line
41		ndmfv 6434	. . 3 ⊢ (¬ ⟨𝐴, 𝐴⟩ ∈ dom Line → (Line‘⟨𝐴, 𝐴⟩) = ∅)
42	40, 41	ax-mp 5	. 2 ⊢ (Line‘⟨𝐴, 𝐴⟩) = ∅
43	1, 42	eqtri 2828	1 ⊢ (𝐴Line𝐴) = ∅

Syntax hints:  ¬ wn 3   ↔ wb 197   ∧ wa 384   ∧ w3a 1100   = wceq 1637  ∃wex 1859   ∈ wcel 2156   ≠ wne 2978  ∃wrex 3097  Vcvv 3391  ∅c0 4116  ⟨cop 4376  {copab 4906   × cxp 5309  ^◡ccnv 5310  dom cdm 5311  ‘cfv 6097  (class class class)co 6870  {coprab 6871  [cec 7973  ℕcn 11301  𝔼cee 25978   Colinear ccolin 32460  Linecline2 32557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-xp 5317  df-cnv 5319  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fv 6105  df-ov 6873  df-oprab 6874  df-ec 7977  df-line2 32560

This theorem is referenced by: (None)