Theorem ncolne1 28711

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 2737	. . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺)
14	2, 13, 4, 6, 12, 10	tgbtwntriv1 28577	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑋𝐼𝑍))
15		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
16	15	oveq1d 7377	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋𝐼𝑍) = (𝑌𝐼𝑍))
17	14, 16	eleqtrd 2839	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑌𝐼𝑍))
18	2, 3, 4, 6, 8, 10, 12, 17	btwncolg1 28641	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
19	1, 18	mtand 816	. 2 ⊢ (𝜑 → ¬ 𝑋 = 𝑌)
20	19	neqned 2940	1 ⊢ (𝜑 → 𝑋 ≠ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ‘cfv 6494  (class class class)co 7362  Basecbs 17174  distcds 17224  TarskiGcstrkg 28513  Itvcitv 28519  LineGclng 28520

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 5372

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 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-iota 6450  df-fun 6496  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-trkgc 28534  df-trkgb 28535  df-trkgcb 28536  df-trkg 28539

This theorem is referenced by:  ncolne2  28712  tglineneq  28730  midexlem  28778  mideulem2  28820  outpasch  28841  hlpasch  28842  trgcopy  28890  trgcopyeulem  28891  acopy  28919  acopyeu  28920  cgrg3col4  28939  tgasa1  28944