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Theorem lpni 28263
Description: For any line in a planar incidence geometry, there exists a point not on the line. (Contributed by Jeff Hankins, 15-Aug-2009.)
Hypothesis
Ref Expression
l2p.1 𝑃 = 𝐺
Assertion
Ref Expression
lpni ((𝐺 ∈ Plig ∧ 𝐿𝐺) → ∃𝑎𝑃 𝑎𝐿)
Distinct variable groups:   𝐺,𝑎   𝐿,𝑎   𝑃,𝑎

Proof of Theorem lpni
Dummy variables 𝑏 𝑐 𝑙 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 l2p.1 . . . 4 𝑃 = 𝐺
21tncp 28261 . . 3 (𝐺 ∈ Plig → ∃𝑏𝑃𝑐𝑃𝑑𝑃𝑙𝐺 ¬ (𝑏𝑙𝑐𝑙𝑑𝑙))
3 eleq2 2878 . . . . . . . . . 10 (𝑙 = 𝐿 → (𝑏𝑙𝑏𝐿))
4 eleq2 2878 . . . . . . . . . 10 (𝑙 = 𝐿 → (𝑐𝑙𝑐𝐿))
5 eleq2 2878 . . . . . . . . . 10 (𝑙 = 𝐿 → (𝑑𝑙𝑑𝐿))
63, 4, 53anbi123d 1433 . . . . . . . . 9 (𝑙 = 𝐿 → ((𝑏𝑙𝑐𝑙𝑑𝑙) ↔ (𝑏𝐿𝑐𝐿𝑑𝐿)))
76notbid 321 . . . . . . . 8 (𝑙 = 𝐿 → (¬ (𝑏𝑙𝑐𝑙𝑑𝑙) ↔ ¬ (𝑏𝐿𝑐𝐿𝑑𝐿)))
87rspccv 3568 . . . . . . 7 (∀𝑙𝐺 ¬ (𝑏𝑙𝑐𝑙𝑑𝑙) → (𝐿𝐺 → ¬ (𝑏𝐿𝑐𝐿𝑑𝐿)))
9 eleq1w 2872 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝑎𝐿𝑏𝐿))
109notbid 321 . . . . . . . . . . 11 (𝑎 = 𝑏 → (¬ 𝑎𝐿 ↔ ¬ 𝑏𝐿))
1110rspcev 3571 . . . . . . . . . 10 ((𝑏𝑃 ∧ ¬ 𝑏𝐿) → ∃𝑎𝑃 ¬ 𝑎𝐿)
1211ex 416 . . . . . . . . 9 (𝑏𝑃 → (¬ 𝑏𝐿 → ∃𝑎𝑃 ¬ 𝑎𝐿))
13 eleq1w 2872 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (𝑎𝐿𝑐𝐿))
1413notbid 321 . . . . . . . . . . 11 (𝑎 = 𝑐 → (¬ 𝑎𝐿 ↔ ¬ 𝑐𝐿))
1514rspcev 3571 . . . . . . . . . 10 ((𝑐𝑃 ∧ ¬ 𝑐𝐿) → ∃𝑎𝑃 ¬ 𝑎𝐿)
1615ex 416 . . . . . . . . 9 (𝑐𝑃 → (¬ 𝑐𝐿 → ∃𝑎𝑃 ¬ 𝑎𝐿))
17 eleq1w 2872 . . . . . . . . . . . 12 (𝑎 = 𝑑 → (𝑎𝐿𝑑𝐿))
1817notbid 321 . . . . . . . . . . 11 (𝑎 = 𝑑 → (¬ 𝑎𝐿 ↔ ¬ 𝑑𝐿))
1918rspcev 3571 . . . . . . . . . 10 ((𝑑𝑃 ∧ ¬ 𝑑𝐿) → ∃𝑎𝑃 ¬ 𝑎𝐿)
2019ex 416 . . . . . . . . 9 (𝑑𝑃 → (¬ 𝑑𝐿 → ∃𝑎𝑃 ¬ 𝑎𝐿))
2112, 16, 203jaao 1430 . . . . . . . 8 ((𝑏𝑃𝑐𝑃𝑑𝑃) → ((¬ 𝑏𝐿 ∨ ¬ 𝑐𝐿 ∨ ¬ 𝑑𝐿) → ∃𝑎𝑃 ¬ 𝑎𝐿))
22 3ianor 1104 . . . . . . . 8 (¬ (𝑏𝐿𝑐𝐿𝑑𝐿) ↔ (¬ 𝑏𝐿 ∨ ¬ 𝑐𝐿 ∨ ¬ 𝑑𝐿))
23 df-nel 3092 . . . . . . . . 9 (𝑎𝐿 ↔ ¬ 𝑎𝐿)
2423rexbii 3210 . . . . . . . 8 (∃𝑎𝑃 𝑎𝐿 ↔ ∃𝑎𝑃 ¬ 𝑎𝐿)
2521, 22, 243imtr4g 299 . . . . . . 7 ((𝑏𝑃𝑐𝑃𝑑𝑃) → (¬ (𝑏𝐿𝑐𝐿𝑑𝐿) → ∃𝑎𝑃 𝑎𝐿))
268, 25syl9r 78 . . . . . 6 ((𝑏𝑃𝑐𝑃𝑑𝑃) → (∀𝑙𝐺 ¬ (𝑏𝑙𝑐𝑙𝑑𝑙) → (𝐿𝐺 → ∃𝑎𝑃 𝑎𝐿)))
27263expia 1118 . . . . 5 ((𝑏𝑃𝑐𝑃) → (𝑑𝑃 → (∀𝑙𝐺 ¬ (𝑏𝑙𝑐𝑙𝑑𝑙) → (𝐿𝐺 → ∃𝑎𝑃 𝑎𝐿))))
2827rexlimdv 3242 . . . 4 ((𝑏𝑃𝑐𝑃) → (∃𝑑𝑃𝑙𝐺 ¬ (𝑏𝑙𝑐𝑙𝑑𝑙) → (𝐿𝐺 → ∃𝑎𝑃 𝑎𝐿)))
2928rexlimivv 3251 . . 3 (∃𝑏𝑃𝑐𝑃𝑑𝑃𝑙𝐺 ¬ (𝑏𝑙𝑐𝑙𝑑𝑙) → (𝐿𝐺 → ∃𝑎𝑃 𝑎𝐿))
302, 29syl 17 . 2 (𝐺 ∈ Plig → (𝐿𝐺 → ∃𝑎𝑃 𝑎𝐿))
3130imp 410 1 ((𝐺 ∈ Plig ∧ 𝐿𝐺) → ∃𝑎𝑃 𝑎𝐿)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3o 1083  w3a 1084   = wceq 1538  wcel 2111  wnel 3091  wral 3106  wrex 3107   cuni 4800  Pligcplig 28257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-v 3443  df-in 3888  df-ss 3898  df-uni 4801  df-plig 28258
This theorem is referenced by: (None)
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