Theorem nsnlplig 30770

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.

Step	Hyp	Ref	Expression
1		eqid 2769	. . . 4 ⊢ ∪ 𝐺 = ∪ 𝐺
2	1	l2p 30768	. . 3 ⊢ ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → ∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}))
3		elsni 4608	. . . . . . . 8 ⊢ (𝑎 ∈ {𝐴} → 𝑎 = 𝐴)
4		elsni 4608	. . . . . . . 8 ⊢ (𝑏 ∈ {𝐴} → 𝑏 = 𝐴)
5		eqtr3 2791	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐴) → 𝑎 = 𝑏)
6		eqneqall 2975	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑎 ≠ 𝑏 → ¬ {𝐴} ∈ 𝐺))
7	5, 6	syl 18	. . . . . . . 8 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐴) → (𝑎 ≠ 𝑏 → ¬ {𝐴} ∈ 𝐺))
8	3, 4, 7	syl2an 607	. . . . . . 7 ⊢ ((𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → (𝑎 ≠ 𝑏 → ¬ {𝐴} ∈ 𝐺))
9	8	impcom 412	. . . . . 6 ⊢ ((𝑎 ≠ 𝑏 ∧ (𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴})) → ¬ {𝐴} ∈ 𝐺)
10	9	3impb 1130	. . . . 5 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → ¬ {𝐴} ∈ 𝐺)
11	10	a1i 11	. . . 4 ⊢ ((𝑎 ∈ ∪ 𝐺 ∧ 𝑏 ∈ ∪ 𝐺) → ((𝑎 ≠ 𝑏 ∧ 𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → ¬ {𝐴} ∈ 𝐺))
12	11	rexlimivv 3213	. . 3 ⊢ (∃𝑎 ∈ ∪ 𝐺∃𝑏 ∈ ∪ 𝐺(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ {𝐴} ∧ 𝑏 ∈ {𝐴}) → ¬ {𝐴} ∈ 𝐺)
13	2, 12	syl 18	. 2 ⊢ ((𝐺 ∈ Plig ∧ {𝐴} ∈ 𝐺) → ¬ {𝐴} ∈ 𝐺)
14	13	pm2.01da 810	1 ⊢ (𝐺 ∈ Plig → ¬ {𝐴} ∈ 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095  {csn 4591  ∪ cuni 4873  Pligcplig 30763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-v 3465  df-ss 3930  df-sn 4592  df-uni 4874  df-plig 30764

This theorem is referenced by:  n0lplig  30772