Theorem ncolne1 28552

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 2729	. . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺)
14	2, 13, 4, 6, 12, 10	tgbtwntriv1 28418	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑋𝐼𝑍))
15		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
16	15	oveq1d 7402	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋𝐼𝑍) = (𝑌𝐼𝑍))
17	14, 16	eleqtrd 2830	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑌𝐼𝑍))
18	2, 3, 4, 6, 8, 10, 12, 17	btwncolg1 28482	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
19	1, 18	mtand 815	. 2 ⊢ (𝜑 → ¬ 𝑋 = 𝑌)
20	19	neqned 2932	1 ⊢ (𝜑 → 𝑋 ≠ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  distcds 17229  TarskiGcstrkg 28354  Itvcitv 28360  LineGclng 28361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-trkgc 28375  df-trkgb 28376  df-trkgcb 28377  df-trkg 28380

This theorem is referenced by:  ncolne2  28553  tglineneq  28571  midexlem  28619  mideulem2  28661  outpasch  28682  hlpasch  28683  trgcopy  28731  trgcopyeulem  28732  acopy  28760  acopyeu  28761  cgrg3col4  28780  tgasa1  28785