Theorem ncolne2 28711

Description: Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) TODO (NM): maybe ncolne2 28711 could be simplified out and deleted, replaced by ncolcom 28646.

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 28646	. 2 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))
10	1, 2, 3, 4, 5, 6, 7, 9	ncolne1 28710	1 ⊢ (𝜑 → 𝑋 ≠ 𝑍)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  TarskiGcstrkg 28512  Itvcitv 28518  LineGclng 28519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-trkgc 28533  df-trkgb 28534  df-trkgcb 28535  df-trkg 28538

This theorem is referenced by:  midexlem  28777