Theorem ncolne2 29106

Description: Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) TODO (NM): maybe ncolne2 29106 could be simplified out and deleted, replaced by ncolcom 29036.

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 29036	. 2 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))
10	1, 2, 3, 4, 5, 6, 7, 9	ncolne1 29105	1 ⊢ (𝜑 → 𝑋 ≠ 𝑍)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ wo 861   = wceq 1569   ∈ wcel 2146   ≠ wne 2958  ‘cfv 6546  (class class class)co 7427  Basecbs 17399  TarskiGcstrkg 28901  Itvcitv 28907  LineGclng 28908

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  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 3416  df-v 3455  df-sbc 3742  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6500  df-fun 6548  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-trkgc 28922  df-trkgb 28923  df-trkgcb 28924  df-trkg 28927

This theorem is used by:  midexlem  29176  symquadmid  29316  prlngsymquadopp  29455  quadcgrprlng  29456  tgextamirn  33079  tgextahpl  33081  tgextamirnhl  33084  tgextancgra  33086  tgextalt1  33088  tgextalt2  33089  tgexta  33090  tgaasexc  33093