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Theorem l2p 30201
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 30198 . . . 4 (𝐺 ∈ Plig → (𝐺 ∈ Plig ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
3 eleq2 2814 . . . . . . . 8 (𝑙 = 𝐿 → (𝑎𝑙𝑎𝐿))
4 eleq2 2814 . . . . . . . 8 (𝑙 = 𝐿 → (𝑏𝑙𝑏𝐿))
53, 43anbi23d 1435 . . . . . . 7 (𝑙 = 𝐿 → ((𝑎𝑏𝑎𝑙𝑏𝑙) ↔ (𝑎𝑏𝑎𝐿𝑏𝐿)))
652rexbidv 3211 . . . . . 6 (𝑙 = 𝐿 → (∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
76rspccv 3601 . . . . 5 (∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
873ad2ant2 1131 . . . 4 ((∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)) → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
92, 8syl6bi 253 . . 3 (𝐺 ∈ Plig → (𝐺 ∈ Plig → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿))))
109pm2.43i 52 . 2 (𝐺 ∈ Plig → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
1110imp 406 1 ((𝐺 ∈ Plig ∧ 𝐿𝐺) → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wne 2932  wral 3053  wrex 3062  ∃!wreu 3366   cuni 4899  Pligcplig 30196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-v 3468  df-in 3947  df-ss 3957  df-uni 4900  df-plig 30197
This theorem is referenced by:  nsnlplig  30203  nsnlpligALT  30204  n0lpligALT  30206
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