Theorem ncolne1 28907

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 485	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝐺 ∈ TarskiG)
7		ncolne.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8	7	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑌 ∈ 𝐵)
9		ncolne.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
10	9	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ 𝐵)
11		ncolne.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	11	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝐵)
13		eqid 2763	. . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺)
14	2, 13, 4, 6, 12, 10	tgbtwntriv1 28769	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑋𝐼𝑍))
15		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
16	15	oveq1d 7425	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋𝐼𝑍) = (𝑌𝐼𝑍))
17	14, 16	eleqtrd 2865	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑌𝐼𝑍))
18	2, 3, 4, 6, 8, 10, 12, 17	btwncolg1 28833	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
19	1, 18	mtand 827	. 2 ⊢ (𝜑 → ¬ 𝑋 = 𝑌)
20	19	neqned 2965	1 ⊢ (𝜑 → 𝑋 ≠ 𝑌)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ‘cfv 6536  (class class class)co 7410  Basecbs 17273  distcds 17323  TarskiGcstrkg 28705  Itvcitv 28711  LineGclng 28712

This proof depends on 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 proof 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 28726  df-trkgb 28727  df-trkgcb 28728  df-trkg 28731

This theorem is used by:  ncolne2  28908  tglineneq  28927  midexlem  28978  mideulem2  29024  outpasch  29046  hlpasch  29047  trgcopy  29124  trgcopyeulem  29125  acopy  29153  acopyeu  29154  cgrg3col4  29179  tgasa1  29184  symquadprlng  29221  prlngsymquadlem  29222  quadcgrprlng  29225