Theorem tgbtwnne 28566

Description: Betweenness and inequality. (Contributed by Thierry Arnoux, 1-Dec-2019.)

Hypotheses

Ref	Expression
tkgeom.p	⊢ 𝑃 = (Base‘𝐺)
tkgeom.d	⊢ − = (dist‘𝐺)
tkgeom.i	⊢ 𝐼 = (Itv‘𝐺)
tkgeom.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgbtwntriv2.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgbtwntriv2.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgbtwncomb.3	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgbtwnne.1	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnne.2	⊢ (𝜑 → 𝐵 ≠ 𝐴)

Assertion

Ref	Expression
tgbtwnne	⊢ (𝜑 → 𝐴 ≠ 𝐶)

Proof of Theorem tgbtwnne

Step	Hyp	Ref	Expression
1		tkgeom.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		tkgeom.d	. . . . 5 ⊢ − = (dist‘𝐺)
3		tkgeom.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
4		tkgeom.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
6		tgbtwntriv2.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ 𝑃)
8		tgbtwntriv2.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ∈ 𝑃)
10		tgbtwnne.1	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
11	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
12		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶)
13	12	oveq2d 7376	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴𝐼𝐴) = (𝐴𝐼𝐶))
14	11, 13	eleqtrrd 2840	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐴))
15	1, 2, 3, 5, 7, 9, 14	axtgbtwnid 28542	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐵)
16	15	eqcomd 2743	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐴)
17		tgbtwnne.2	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
18	17	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ≠ 𝐴)
19	18	neneqd 2938	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → ¬ 𝐵 = 𝐴)
20	16, 19	pm2.65da 817	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐶)
21	20	neqned 2940	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ‘cfv 6493  (class class class)co 7360  Basecbs 17140  distcds 17190  TarskiGcstrkg 28503  Itvcitv 28509

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-ext 2709  ax-nul 5252

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6449  df-fv 6501  df-ov 7363  df-trkgb 28525  df-trkg 28529

This theorem is referenced by:  mideulem2  28810  opphllem  28811  outpasch  28831  lnopp2hpgb  28839  lmieu  28860  dfcgra2  28906