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Theorem nsnlplig 27880
 Description: There is no "one-point line" in a planar incidence geometry. (Contributed by BJ, 2-Dec-2021.) (Proof shortened by AV, 5-Dec-2021.)
Assertion
Ref Expression
nsnlplig (𝐺 ∈ Plig → ¬ {𝐴} ∈ 𝐺)

Proof of Theorem nsnlplig
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2825 . . . 4 𝐺 = 𝐺
21l2p 27878 . . 3 ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → ∃𝑎 𝐺𝑏 𝐺(𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}))
3 elsni 4414 . . . . . . . 8 (𝑎 ∈ {𝐴} → 𝑎 = 𝐴)
4 elsni 4414 . . . . . . . 8 (𝑏 ∈ {𝐴} → 𝑏 = 𝐴)
5 eqtr3 2848 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = 𝐴) → 𝑎 = 𝑏)
6 eqneqall 3010 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑎𝑏 → ¬ {𝐴} ∈ 𝐺))
75, 6syl 17 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = 𝐴) → (𝑎𝑏 → ¬ {𝐴} ∈ 𝐺))
83, 4, 7syl2an 589 . . . . . . 7 ((𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → (𝑎𝑏 → ¬ {𝐴} ∈ 𝐺))
98impcom 398 . . . . . 6 ((𝑎𝑏 ∧ (𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴})) → ¬ {𝐴} ∈ 𝐺)
1093impb 1147 . . . . 5 ((𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → ¬ {𝐴} ∈ 𝐺)
1110a1i 11 . . . 4 ((𝑎 𝐺𝑏 𝐺) → ((𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → ¬ {𝐴} ∈ 𝐺))
1211rexlimivv 3246 . . 3 (∃𝑎 𝐺𝑏 𝐺(𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → ¬ {𝐴} ∈ 𝐺)
132, 12syl 17 . 2 ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → ¬ {𝐴} ∈ 𝐺)
1413pm2.01da 833 1 (𝐺 ∈ Plig → ¬ {𝐴} ∈ 𝐺)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  ∃wrex 3118  {csn 4397  ∪ cuni 4658  Pligcplig 27873 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-v 3416  df-sn 4398  df-uni 4659  df-plig 27874 This theorem is referenced by:  n0lplig  27882
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