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Theorem l2p 29307
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 29304 . . . 4 (𝐺 ∈ Plig → (𝐺 ∈ Plig ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
3 eleq2 2826 . . . . . . . 8 (𝑙 = 𝐿 → (𝑎𝑙𝑎𝐿))
4 eleq2 2826 . . . . . . . 8 (𝑙 = 𝐿 → (𝑏𝑙𝑏𝐿))
53, 43anbi23d 1439 . . . . . . 7 (𝑙 = 𝐿 → ((𝑎𝑏𝑎𝑙𝑏𝑙) ↔ (𝑎𝑏𝑎𝐿𝑏𝐿)))
652rexbidv 3211 . . . . . 6 (𝑙 = 𝐿 → (∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ↔ ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
76rspccv 3576 . . . . 5 (∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
873ad2ant2 1134 . . . 4 ((∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)) → (𝐿𝐺 → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿)))
92, 8syl6bi 252 . . 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 1087   = wceq 1541  wcel 2106  wne 2941  wral 3062  wrex 3071  ∃!wreu 3349   cuni 4863  Pligcplig 29302
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-reu 3352  df-v 3445  df-in 3915  df-ss 3925  df-uni 4864  df-plig 29303
This theorem is referenced by:  nsnlplig  29309  nsnlpligALT  29310  n0lpligALT  29312
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