Theorem ncolne2 28649

Description: Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) TODO (NM): maybe ncolne2 28649 could be simplified out and deleted, replaced by ncolcom 28584.

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 28584	. 2 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))
10	1, 2, 3, 4, 5, 6, 7, 9	ncolne1 28648	1 ⊢ (𝜑 → 𝑋 ≠ 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 847   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  TarskiGcstrkg 28450  Itvcitv 28456  LineGclng 28457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-trkgc 28471  df-trkgb 28472  df-trkgcb 28473  df-trkg 28476

This theorem is referenced by:  midexlem  28715