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Theorem l2p 28183
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 28180 . . . 4 (𝐺 ∈ Plig → (𝐺 ∈ Plig ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
3 eleq2 2898 . . . . . . . 8 (𝑙 = 𝐿 → (𝑎𝑙𝑎𝐿))
4 eleq2 2898 . . . . . . . 8 (𝑙 = 𝐿 → (𝑏𝑙𝑏𝐿))
53, 43anbi23d 1430 . . . . . . 7 (𝑙 = 𝐿 → ((𝑎𝑏𝑎𝑙𝑏𝑙) ↔ (𝑎𝑏𝑎𝐿𝑏𝐿)))
652rexbidv 3297 . . . . . 6 (𝑙 = 𝐿 → (∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
76rspccv 3617 . . . . 5 (∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
873ad2ant2 1126 . . . 4 ((∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)) → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
92, 8syl6bi 254 . . 3 (𝐺 ∈ Plig → (𝐺 ∈ Plig → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿))))
109pm2.43i 52 . 2 (𝐺 ∈ Plig → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
1110imp 407 1 ((𝐺 ∈ Plig ∧ 𝐿𝐺) → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  ∃!wreu 3137   cuni 4830  Pligcplig 28178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-reu 3142  df-uni 4831  df-plig 28179
This theorem is referenced by:  nsnlplig  28185  nsnlpligALT  28186  n0lpligALT  28188
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