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Theorem l2p 28265
Description: For any line in a planar incidence geometry, there exist two different points on the line. (Contributed by AV, 28-Nov-2021.)
Hypothesis
Ref Expression
l2p.1 𝑃 = 𝐺
Assertion
Ref Expression
l2p ((𝐺 ∈ Plig ∧ 𝐿𝐺) → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿))
Distinct variable groups:   𝑎,𝑏,𝐺   𝐿,𝑎,𝑏   𝑃,𝑎,𝑏

Proof of Theorem l2p
Dummy variables 𝑐 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 l2p.1 . . . . 5 𝑃 = 𝐺
21isplig 28262 . . . 4 (𝐺 ∈ Plig → (𝐺 ∈ Plig ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
3 eleq2 2881 . . . . . . . 8 (𝑙 = 𝐿 → (𝑎𝑙𝑎𝐿))
4 eleq2 2881 . . . . . . . 8 (𝑙 = 𝐿 → (𝑏𝑙𝑏𝐿))
53, 43anbi23d 1436 . . . . . . 7 (𝑙 = 𝐿 → ((𝑎𝑏𝑎𝑙𝑏𝑙) ↔ (𝑎𝑏𝑎𝐿𝑏𝐿)))
652rexbidv 3262 . . . . . 6 (𝑙 = 𝐿 → (∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
76rspccv 3571 . . . . 5 (∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
873ad2ant2 1131 . . . 4 ((∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)) → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
92, 8syl6bi 256 . . 3 (𝐺 ∈ Plig → (𝐺 ∈ Plig → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿))))
109pm2.43i 52 . 2 (𝐺 ∈ Plig → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
1110imp 410 1 ((𝐺 ∈ Plig ∧ 𝐿𝐺) → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110  ∃!wreu 3111   cuni 4803  Pligcplig 28260
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-reu 3116  df-v 3446  df-in 3891  df-ss 3901  df-uni 4804  df-plig 28261
This theorem is referenced by:  nsnlplig  28267  nsnlpligALT  28268  n0lpligALT  28270
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