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Theorem tncp 26721
 Description: In any planar incidence geometry, there exist three non-collinear points. (Contributed by FL, 3-Aug-2009.)
Hypothesis
Ref Expression
tncp.1 𝑃 = 𝐿
Assertion
Ref Expression
tncp (𝐿 ∈ Plig → ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))
Distinct variable groups:   𝐿,𝑎,𝑏,𝑐,𝑙   𝑃,𝑎,𝑏,𝑐
Allowed substitution hint:   𝑃(𝑙)

Proof of Theorem tncp
StepHypRef Expression
1 tncp.1 . . . 4 𝑃 = 𝐿
21isplig 26720 . . 3 (𝐿 ∈ Plig → (𝐿 ∈ Plig ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐿 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐿𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
32ibi 255 . 2 (𝐿 ∈ Plig → (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐿 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐿𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙)))
43simp3d 1068 1 (𝐿 ∈ Plig → ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  ∪ cuni 4372  Pligcplig 26718 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-v 3175  df-uni 4373  df-plig 26719 This theorem is referenced by:  lpni  26722
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