MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nsnlplig Structured version   Visualization version   GIF version

Theorem nsnlplig 28185
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 2818 . . . 4 𝐺 = 𝐺
21l2p 28183 . . 3 ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → ∃𝑎 𝐺𝑏 𝐺(𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}))
3 elsni 4574 . . . . . . . 8 (𝑎 ∈ {𝐴} → 𝑎 = 𝐴)
4 elsni 4574 . . . . . . . 8 (𝑏 ∈ {𝐴} → 𝑏 = 𝐴)
5 eqtr3 2840 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = 𝐴) → 𝑎 = 𝑏)
6 eqneqall 3024 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑎𝑏 → ¬ {𝐴} ∈ 𝐺))
75, 6syl 17 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = 𝐴) → (𝑎𝑏 → ¬ {𝐴} ∈ 𝐺))
83, 4, 7syl2an 595 . . . . . . 7 ((𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → (𝑎𝑏 → ¬ {𝐴} ∈ 𝐺))
98impcom 408 . . . . . 6 ((𝑎𝑏 ∧ (𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴})) → ¬ {𝐴} ∈ 𝐺)
1093impb 1107 . . . . 5 ((𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → ¬ {𝐴} ∈ 𝐺)
1110a1i 11 . . . 4 ((𝑎 𝐺𝑏 𝐺) → ((𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → ¬ {𝐴} ∈ 𝐺))
1211rexlimivv 3289 . . 3 (∃𝑎 𝐺𝑏 𝐺(𝑎𝑏𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → ¬ {𝐴} ∈ 𝐺)
132, 12syl 17 . 2 ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → ¬ {𝐴} ∈ 𝐺)
1413pm2.01da 795 1 (𝐺 ∈ Plig → ¬ {𝐴} ∈ 𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wrex 3136  {csn 4557   cuni 4830  Pligcplig 28178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-sn 4558  df-uni 4831  df-plig 28179
This theorem is referenced by:  n0lplig  28187
  Copyright terms: Public domain W3C validator