Theorem axtgbtwnid 28943

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 28930	. . . . 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 4136	. . . . . 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 4137	. . . . . 6 ⊢ (TarskiGC ∩ TarskiGB) ⊆ TarskiGB
4	2, 3	sstri 3922	. . . . 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 3959	. . . 4 ⊢ TarskiG ⊆ TarskiGB
6		axtrkg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sselid 3911	. . 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 28932	. . . . 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 503	. . . 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 1160	. . 3 ⊢ (𝐺 ∈ TarskiGB → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦))
14	7, 13	syl 18	. 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 23	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
19	18, 18	oveq12d 7448	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥𝐼𝑥) = (𝑋𝐼𝑋))
20	19	eleq2d 2850	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑥) ↔ 𝑦 ∈ (𝑋𝐼𝑋)))
21		eqeq1 2767	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
22	20, 21	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ↔ (𝑦 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑦)))
23		eleq1 2852	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑋) ↔ 𝑌 ∈ (𝑋𝐼𝑋)))
24		eqeq2 2775	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 = 𝑦 ↔ 𝑋 = 𝑌))
25	23, 24	imbi12d 347	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑦 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑦) ↔ (𝑌 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑌)))
26	22, 25	rspc2v 3590	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) → (𝑌 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑌)))
27	16, 17, 26	syl2anc 596	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) → (𝑌 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑌)))
28	14, 15, 27	mp2d 50	1 ⊢ (𝜑 → 𝑋 = 𝑌)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ w3o 1102   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  {cab 2741  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454  [wsbc 3742   ∖ cdif 4048   ∩ cin 4092  𝒫 cpw 4560  {csn 4588  ‘cfv 6547  (class class class)co 7430   ∈ cmpo 7432  Basecbs 17402  distcds 17452  TarskiGcstrkg 28904  TarskiG_Ccstrkgc 28905  TarskiG_Bcstrkgb 28906  TarskiG_CBcstrkgcb 28907  Itvcitv 28910  LineGclng 28911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6501  df-fv 6555  df-ov 7433  df-trkgb 28926  df-trkg 28930

This theorem is used by:  tgbtwncom  28966  tgbtwnne  28968  tgbtwnswapid  28970  tgbtwnintr  28971  tgifscgr  28986  tgidinside  29049  tgbtwnconn1lem3  29052  coltr3  29132  tglnpt3  29137  mirinv  29153  miriso  29157  krippenlem  29177  midexlem  29179  colperpexlem3  29223  oppne3  29234  oppnid  29237  opphllem1  29238  hlpasch  29249  midid  29301  lmiisolem  29316  tgaaddcpbl  29367  angmgmaddeu3  29396  angmgmaddov2lem  29402  prlngmolem1  29445  f1otrg  29463