Theorem ncolne2 28899

Description: Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) TODO (NM): maybe ncolne2 28899 could be simplified out and deleted, replaced by ncolcom 28830.

Hypotheses

Ref	Expression
tglineelsb2.p	⊢ 𝐵 = (Base‘𝐺)
tglineelsb2.i	⊢ 𝐼 = (Itv‘𝐺)
tglineelsb2.l	⊢ 𝐿 = (LineG‘𝐺)
tglineelsb2.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
ncolne.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ncolne.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
ncolne.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
ncolne.2	⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))

Assertion

Ref	Expression
ncolne2	⊢ (𝜑 → 𝑋 ≠ 𝑍)

Proof of Theorem ncolne2

Step	Hyp	Ref	Expression
1		tglineelsb2.p	. 2 ⊢ 𝐵 = (Base‘𝐺)
2		tglineelsb2.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
3		tglineelsb2.l	. 2 ⊢ 𝐿 = (LineG‘𝐺)
4		tglineelsb2.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		ncolne.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		ncolne.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
7		ncolne.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8		ncolne.2	. . 3 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
9	1, 3, 2, 4, 7, 6, 5, 8	ncolcom 28830	. 2 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))
10	1, 2, 3, 4, 5, 6, 7, 9	ncolne1 28898	1 ⊢ (𝜑 → 𝑋 ≠ 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  TarskiGcstrkg 28696  Itvcitv 28702  LineGclng 28703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-trkgc 28717  df-trkgb 28718  df-trkgcb 28719  df-trkg 28722

This theorem is referenced by:  midexlem  28969  symquadmid  29108  prlngsymquadopp  29215  quadcgrprlng  29216