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Theorem tgbtwnne 28008
Description: Betweenness and inequality. (Contributed by Thierry Arnoux, 1-Dec-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Baseβ€˜πΊ)
tkgeom.d βˆ’ = (distβ€˜πΊ)
tkgeom.i 𝐼 = (Itvβ€˜πΊ)
tkgeom.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
tgbtwntriv2.1 (πœ‘ β†’ 𝐴 ∈ 𝑃)
tgbtwntriv2.2 (πœ‘ β†’ 𝐡 ∈ 𝑃)
tgbtwncomb.3 (πœ‘ β†’ 𝐢 ∈ 𝑃)
tgbtwnne.1 (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐢))
tgbtwnne.2 (πœ‘ β†’ 𝐡 β‰  𝐴)
Assertion
Ref Expression
tgbtwnne (πœ‘ β†’ 𝐴 β‰  𝐢)

Proof of Theorem tgbtwnne
StepHypRef Expression
1 tkgeom.p . . . . 5 𝑃 = (Baseβ€˜πΊ)
2 tkgeom.d . . . . 5 βˆ’ = (distβ€˜πΊ)
3 tkgeom.i . . . . 5 𝐼 = (Itvβ€˜πΊ)
4 tkgeom.g . . . . . 6 (πœ‘ β†’ 𝐺 ∈ TarskiG)
54adantr 479 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝐢) β†’ 𝐺 ∈ TarskiG)
6 tgbtwntriv2.1 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ 𝑃)
76adantr 479 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝐢) β†’ 𝐴 ∈ 𝑃)
8 tgbtwntriv2.2 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ 𝑃)
98adantr 479 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝐢) β†’ 𝐡 ∈ 𝑃)
10 tgbtwnne.1 . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐢))
1110adantr 479 . . . . . 6 ((πœ‘ ∧ 𝐴 = 𝐢) β†’ 𝐡 ∈ (𝐴𝐼𝐢))
12 simpr 483 . . . . . . 7 ((πœ‘ ∧ 𝐴 = 𝐢) β†’ 𝐴 = 𝐢)
1312oveq2d 7427 . . . . . 6 ((πœ‘ ∧ 𝐴 = 𝐢) β†’ (𝐴𝐼𝐴) = (𝐴𝐼𝐢))
1411, 13eleqtrrd 2834 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝐢) β†’ 𝐡 ∈ (𝐴𝐼𝐴))
151, 2, 3, 5, 7, 9, 14axtgbtwnid 27984 . . . 4 ((πœ‘ ∧ 𝐴 = 𝐢) β†’ 𝐴 = 𝐡)
1615eqcomd 2736 . . 3 ((πœ‘ ∧ 𝐴 = 𝐢) β†’ 𝐡 = 𝐴)
17 tgbtwnne.2 . . . . 5 (πœ‘ β†’ 𝐡 β‰  𝐴)
1817adantr 479 . . . 4 ((πœ‘ ∧ 𝐴 = 𝐢) β†’ 𝐡 β‰  𝐴)
1918neneqd 2943 . . 3 ((πœ‘ ∧ 𝐴 = 𝐢) β†’ Β¬ 𝐡 = 𝐴)
2016, 19pm2.65da 813 . 2 (πœ‘ β†’ Β¬ 𝐴 = 𝐢)
2120neqned 2945 1 (πœ‘ β†’ 𝐴 β‰  𝐢)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  distcds 17210  TarskiGcstrkg 27945  Itvcitv 27951
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-trkgb 27967  df-trkg 27971
This theorem is referenced by:  mideulem2  28252  opphllem  28253  outpasch  28273  lnopp2hpgb  28281  lmieu  28302  dfcgra2  28348
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