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Theorem lpni 27791
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 27789 . . 3 (𝐺 ∈ Plig → ∃𝑏𝑃𝑐𝑃𝑑𝑃𝑙𝐺 ¬ (𝑏𝑙𝑐𝑙𝑑𝑙))
3 eleq2 2833 . . . . . . . . . 10 (𝑙 = 𝐿 → (𝑏𝑙𝑏𝐿))
4 eleq2 2833 . . . . . . . . . 10 (𝑙 = 𝐿 → (𝑐𝑙𝑐𝐿))
5 eleq2 2833 . . . . . . . . . 10 (𝑙 = 𝐿 → (𝑑𝑙𝑑𝐿))
63, 4, 53anbi123d 1560 . . . . . . . . 9 (𝑙 = 𝐿 → ((𝑏𝑙𝑐𝑙𝑑𝑙) ↔ (𝑏𝐿𝑐𝐿𝑑𝐿)))
76notbid 309 . . . . . . . 8 (𝑙 = 𝐿 → (¬ (𝑏𝑙𝑐𝑙𝑑𝑙) ↔ ¬ (𝑏𝐿𝑐𝐿𝑑𝐿)))
87rspccv 3458 . . . . . . 7 (∀𝑙𝐺 ¬ (𝑏𝑙𝑐𝑙𝑑𝑙) → (𝐿𝐺 → ¬ (𝑏𝐿𝑐𝐿𝑑𝐿)))
9 eleq1w 2827 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝑎𝐿𝑏𝐿))
109notbid 309 . . . . . . . . . . 11 (𝑎 = 𝑏 → (¬ 𝑎𝐿 ↔ ¬ 𝑏𝐿))
1110rspcev 3461 . . . . . . . . . 10 ((𝑏𝑃 ∧ ¬ 𝑏𝐿) → ∃𝑎𝑃 ¬ 𝑎𝐿)
1211ex 401 . . . . . . . . 9 (𝑏𝑃 → (¬ 𝑏𝐿 → ∃𝑎𝑃 ¬ 𝑎𝐿))
13 eleq1w 2827 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (𝑎𝐿𝑐𝐿))
1413notbid 309 . . . . . . . . . . 11 (𝑎 = 𝑐 → (¬ 𝑎𝐿 ↔ ¬ 𝑐𝐿))
1514rspcev 3461 . . . . . . . . . 10 ((𝑐𝑃 ∧ ¬ 𝑐𝐿) → ∃𝑎𝑃 ¬ 𝑎𝐿)
1615ex 401 . . . . . . . . 9 (𝑐𝑃 → (¬ 𝑐𝐿 → ∃𝑎𝑃 ¬ 𝑎𝐿))
17 eleq1w 2827 . . . . . . . . . . . 12 (𝑎 = 𝑑 → (𝑎𝐿𝑑𝐿))
1817notbid 309 . . . . . . . . . . 11 (𝑎 = 𝑑 → (¬ 𝑎𝐿 ↔ ¬ 𝑑𝐿))
1918rspcev 3461 . . . . . . . . . 10 ((𝑑𝑃 ∧ ¬ 𝑑𝐿) → ∃𝑎𝑃 ¬ 𝑎𝐿)
2019ex 401 . . . . . . . . 9 (𝑑𝑃 → (¬ 𝑑𝐿 → ∃𝑎𝑃 ¬ 𝑎𝐿))
2112, 16, 203jaao 1557 . . . . . . . 8 ((𝑏𝑃𝑐𝑃𝑑𝑃) → ((¬ 𝑏𝐿 ∨ ¬ 𝑐𝐿 ∨ ¬ 𝑑𝐿) → ∃𝑎𝑃 ¬ 𝑎𝐿))
22 3ianor 1132 . . . . . . . 8 (¬ (𝑏𝐿𝑐𝐿𝑑𝐿) ↔ (¬ 𝑏𝐿 ∨ ¬ 𝑐𝐿 ∨ ¬ 𝑑𝐿))
23 df-nel 3041 . . . . . . . . 9 (𝑎𝐿 ↔ ¬ 𝑎𝐿)
2423rexbii 3188 . . . . . . . 8 (∃𝑎𝑃 𝑎𝐿 ↔ ∃𝑎𝑃 ¬ 𝑎𝐿)
2521, 22, 243imtr4g 287 . . . . . . 7 ((𝑏𝑃𝑐𝑃𝑑𝑃) → (¬ (𝑏𝐿𝑐𝐿𝑑𝐿) → ∃𝑎𝑃 𝑎𝐿))
268, 25syl9r 78 . . . . . 6 ((𝑏𝑃𝑐𝑃𝑑𝑃) → (∀𝑙𝐺 ¬ (𝑏𝑙𝑐𝑙𝑑𝑙) → (𝐿𝐺 → ∃𝑎𝑃 𝑎𝐿)))
27263expia 1150 . . . . 5 ((𝑏𝑃𝑐𝑃) → (𝑑𝑃 → (∀𝑙𝐺 ¬ (𝑏𝑙𝑐𝑙𝑑𝑙) → (𝐿𝐺 → ∃𝑎𝑃 𝑎𝐿))))
2827rexlimdv 3177 . . . 4 ((𝑏𝑃𝑐𝑃) → (∃𝑑𝑃𝑙𝐺 ¬ (𝑏𝑙𝑐𝑙𝑑𝑙) → (𝐿𝐺 → ∃𝑎𝑃 𝑎𝐿)))
2928rexlimivv 3183 . . 3 (∃𝑏𝑃𝑐𝑃𝑑𝑃𝑙𝐺 ¬ (𝑏𝑙𝑐𝑙𝑑𝑙) → (𝐿𝐺 → ∃𝑎𝑃 𝑎𝐿))
302, 29syl 17 . 2 (𝐺 ∈ Plig → (𝐿𝐺 → ∃𝑎𝑃 𝑎𝐿))
3130imp 395 1 ((𝐺 ∈ Plig ∧ 𝐿𝐺) → ∃𝑎𝑃 𝑎𝐿)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3o 1106  w3a 1107   = wceq 1652  wcel 2155  wnel 3040  wral 3055  wrex 3056   cuni 4594  Pligcplig 27785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-v 3352  df-uni 4595  df-plig 27786
This theorem is referenced by: (None)
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