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Theorem l2p 27667
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 27664 . . . 4 (𝐺 ∈ Plig → (𝐺 ∈ Plig ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
3 eleq2 2838 . . . . . . . 8 (𝑙 = 𝐿 → (𝑎𝑙𝑎𝐿))
4 eleq2 2838 . . . . . . . 8 (𝑙 = 𝐿 → (𝑏𝑙𝑏𝐿))
53, 43anbi23d 1549 . . . . . . 7 (𝑙 = 𝐿 → ((𝑎𝑏𝑎𝑙𝑏𝑙) ↔ (𝑎𝑏𝑎𝐿𝑏𝐿)))
652rexbidv 3204 . . . . . 6 (𝑙 = 𝐿 → (∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
76rspccv 3455 . . . . 5 (∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
873ad2ant2 1127 . . . 4 ((∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)) → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
92, 8syl6bi 243 . . 3 (𝐺 ∈ Plig → (𝐺 ∈ Plig → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿))))
109pm2.43i 52 . 2 (𝐺 ∈ Plig → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
1110imp 393 1 ((𝐺 ∈ Plig ∧ 𝐿𝐺) → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wral 3060  wrex 3061  ∃!wreu 3062   cuni 4572  Pligcplig 27662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-reu 3067  df-v 3351  df-uni 4573  df-plig 27663
This theorem is referenced by:  nsnlplig  27669  nsnlpligALT  27670  n0lpligALT  27672
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