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Theorem ispligb 30496
Description: The predicate "is a planar incidence geometry". (Contributed by BJ, 2-Dec-2021.)
Hypothesis
Ref Expression
isplig.1 𝑃 = 𝐺
Assertion
Ref Expression
ispligb (𝐺 ∈ Plig ↔ (𝐺 ∈ V ∧ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
Distinct variable groups:   𝑎,𝑏,𝑐,𝑙,𝐺   𝑃,𝑎,𝑏,𝑐
Allowed substitution hint:   𝑃(𝑙)

Proof of Theorem ispligb
StepHypRef Expression
1 elex 3501 . 2 (𝐺 ∈ Plig → 𝐺 ∈ V)
2 isplig.1 . . 3 𝑃 = 𝐺
32isplig 30495 . 2 (𝐺 ∈ V → (𝐺 ∈ Plig ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
41, 3biadanii 822 1 (𝐺 ∈ Plig ↔ (𝐺 ∈ V ∧ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  ∃!wreu 3378  Vcvv 3480   cuni 4907  Pligcplig 30493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-v 3482  df-ss 3968  df-uni 4908  df-plig 30494
This theorem is referenced by: (None)
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