Theorem axtgbtwnid 28474

Description: Identity of Betweenness. Axiom A6 of [Schwabhauser] p. 11. (Contributed by Thierry Arnoux, 15-Mar-2019.)

Hypotheses

Ref	Expression
axtrkg.p	⊢ 𝑃 = (Base‘𝐺)
axtrkg.d	⊢ − = (dist‘𝐺)
axtrkg.i	⊢ 𝐼 = (Itv‘𝐺)
axtrkg.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
axtgbtwnid.1	⊢ (𝜑 → 𝑋 ∈ 𝑃)
axtgbtwnid.2	⊢ (𝜑 → 𝑌 ∈ 𝑃)
axtgbtwnid.3	⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑋))

Assertion

Ref	Expression
axtgbtwnid	⊢ (𝜑 → 𝑋 = 𝑌)

Proof of Theorem axtgbtwnid

Dummy variables 𝑓 𝑖 𝑝 𝑥 𝑦 𝑧 𝑎 𝑏 𝑣 𝑠 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-trkg 28461	. . . . 5 ⊢ TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
2		inss1 4237	. . . . . 6 ⊢ ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ (TarskiGC ∩ TarskiGB)
3		inss2 4238	. . . . . 6 ⊢ (TarskiGC ∩ TarskiGB) ⊆ TarskiGB
4	2, 3	sstri 3993	. . . . 5 ⊢ ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ TarskiGB
5	1, 4	eqsstri 4030	. . . 4 ⊢ TarskiG ⊆ TarskiGB
6		axtrkg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sselid 3981	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiGB)
8		axtrkg.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
9		axtrkg.d	. . . . . 6 ⊢ − = (dist‘𝐺)
10		axtrkg.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
11	8, 9, 10	istrkgb 28463	. . . . 5 ⊢ (𝐺 ∈ TarskiGB ↔ (𝐺 ∈ V ∧ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃∀𝑡 ∈ 𝒫 𝑃(∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦)))))
12	11	simprbi 496	. . . 4 ⊢ (𝐺 ∈ TarskiGB → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎 ∈ 𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃∀𝑡 ∈ 𝒫 𝑃(∃𝑎 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏 ∈ 𝑃 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝐼𝑦))))
13	12	simp1d 1143	. . 3 ⊢ (𝐺 ∈ TarskiGB → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦))
14	7, 13	syl 17	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦))
15		axtgbtwnid.3	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑋))
16		axtgbtwnid.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
17		axtgbtwnid.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
18		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
19	18, 18	oveq12d 7449	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥𝐼𝑥) = (𝑋𝐼𝑋))
20	19	eleq2d 2827	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑥) ↔ 𝑦 ∈ (𝑋𝐼𝑋)))
21		eqeq1 2741	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
22	20, 21	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ↔ (𝑦 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑦)))
23		eleq1 2829	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑋) ↔ 𝑌 ∈ (𝑋𝐼𝑋)))
24		eqeq2 2749	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 = 𝑦 ↔ 𝑋 = 𝑌))
25	23, 24	imbi12d 344	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑦 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑦) ↔ (𝑌 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑌)))
26	22, 25	rspc2v 3633	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) → (𝑌 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑌)))
27	16, 17, 26	syl2anc 584	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) → (𝑌 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑌)))
28	14, 15, 27	mp2d 49	1 ⊢ (𝜑 → 𝑋 = 𝑌)

Syntax hints:   → wi 4   ∧ wa 395   ∨ w3o 1086   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480  [wsbc 3788   ∖ cdif 3948   ∩ cin 3950  𝒫 cpw 4600  {csn 4626  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  TarskiG_Ccstrkgc 28436  TarskiG_Bcstrkgb 28437  TarskiG_CBcstrkgcb 28438  Itvcitv 28441  LineGclng 28442

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-trkgb 28457  df-trkg 28461

This theorem is referenced by:  tgbtwncom  28496  tgbtwnne  28498  tgbtwnswapid  28500  tgbtwnintr  28501  tgifscgr  28516  tgidinside  28579  tgbtwnconn1lem3  28582  coltr3  28656  mirinv  28674  miriso  28678  krippenlem  28698  midexlem  28700  colperpexlem3  28740  oppne3  28751  oppnid  28754  opphllem1  28755  hlpasch  28764  midid  28789  lmiisolem  28804  f1otrg  28879