Theorem ncolne1 26967

Description: Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.)

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
ncolne1	⊢ (𝜑 → 𝑋 ≠ 𝑌)

Proof of Theorem ncolne1

Step	Hyp	Ref	Expression
1		ncolne.2	. . 3 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
2		tglineelsb2.p	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3		tglineelsb2.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
4		tglineelsb2.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
5		tglineelsb2.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝐺 ∈ TarskiG)
7		ncolne.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑌 ∈ 𝐵)
9		ncolne.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ 𝐵)
11		ncolne.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	11	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝐵)
13		eqid 2739	. . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺)
14	2, 13, 4, 6, 12, 10	tgbtwntriv1 26833	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑋𝐼𝑍))
15		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
16	15	oveq1d 7283	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋𝐼𝑍) = (𝑌𝐼𝑍))
17	14, 16	eleqtrd 2842	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑌𝐼𝑍))
18	2, 3, 4, 6, 8, 10, 12, 17	btwncolg1 26897	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
19	1, 18	mtand 812	. 2 ⊢ (𝜑 → ¬ 𝑋 = 𝑌)
20	19	neqned 2951	1 ⊢ (𝜑 → 𝑋 ≠ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   = wceq 1541   ∈ wcel 2109   ≠ wne 2944  ‘cfv 6430  (class class class)co 7268  Basecbs 16893  distcds 16952  TarskiGcstrkg 26769  Itvcitv 26775  LineGclng 26776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-iota 6388  df-fun 6432  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-trkgc 26790  df-trkgb 26791  df-trkgcb 26792  df-trkg 26795

This theorem is referenced by:  ncolne2  26968  tglineneq  26986  midexlem  27034  mideulem2  27076  outpasch  27097  hlpasch  27098  trgcopy  27146  trgcopyeulem  27147  acopy  27175  acopyeu  27176  cgrg3col4  27195  tgasa1  27200