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Theorem nsnlplig 28250
 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 2819 . . . 4 𝐺 = 𝐺
21l2p 28248 . . 3 ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → ∃𝑎 𝐺𝑏 𝐺(𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}))
3 elsni 4576 . . . . . . . 8 (𝑎 ∈ {𝐴} → 𝑎 = 𝐴)
4 elsni 4576 . . . . . . . 8 (𝑏 ∈ {𝐴} → 𝑏 = 𝐴)
5 eqtr3 2841 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = 𝐴) → 𝑎 = 𝑏)
6 eqneqall 3025 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑎𝑏 → ¬ {𝐴} ∈ 𝐺))
75, 6syl 17 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = 𝐴) → (𝑎𝑏 → ¬ {𝐴} ∈ 𝐺))
83, 4, 7syl2an 597 . . . . . . 7 ((𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → (𝑎𝑏 → ¬ {𝐴} ∈ 𝐺))
98impcom 410 . . . . . 6 ((𝑎𝑏 ∧ (𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴})) → ¬ {𝐴} ∈ 𝐺)
1093impb 1110 . . . . 5 ((𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → ¬ {𝐴} ∈ 𝐺)
1110a1i 11 . . . 4 ((𝑎 𝐺𝑏 𝐺) → ((𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → ¬ {𝐴} ∈ 𝐺))
1211rexlimivv 3290 . . 3 (∃𝑎 𝐺𝑏 𝐺(𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → ¬ {𝐴} ∈ 𝐺)
132, 12syl 17 . 2 ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → ¬ {𝐴} ∈ 𝐺)
1413pm2.01da 797 1 (𝐺 ∈ Plig → ¬ {𝐴} ∈ 𝐺)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1082   = wceq 1531   ∈ wcel 2108   ≠ wne 3014  ∃wrex 3137  {csn 4559  ∪ cuni 4830  Pligcplig 28243 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-sn 4560  df-uni 4831  df-plig 28244 This theorem is referenced by:  n0lplig  28252
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