Theorem axtgbtwnid 28444

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 28431	. . . . 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 4184	. . . . . 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 4185	. . . . . 6 ⊢ (TarskiGC ∩ TarskiGB) ⊆ TarskiGB
4	2, 3	sstri 3939	. . . . 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 3976	. . . 4 ⊢ TarskiG ⊆ TarskiGB
6		axtrkg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sselid 3927	. . 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 28433	. . . . 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 1142	. . 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 7364	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥𝐼𝑥) = (𝑋𝐼𝑋))
20	19	eleq2d 2817	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑥) ↔ 𝑦 ∈ (𝑋𝐼𝑋)))
21		eqeq1 2735	. . . . 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 2819	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑋) ↔ 𝑌 ∈ (𝑋𝐼𝑋)))
24		eqeq2 2743	. . . . 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 3583	. . 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 1085   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056  {crab 3395  Vcvv 3436  [wsbc 3736   ∖ cdif 3894   ∩ cin 3896  𝒫 cpw 4547  {csn 4573  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  Basecbs 17120  distcds 17170  TarskiGcstrkg 28405  TarskiG_Ccstrkgc 28406  TarskiG_Bcstrkgb 28407  TarskiG_CBcstrkgcb 28408  Itvcitv 28411  LineGclng 28412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-trkgb 28427  df-trkg 28431

This theorem is referenced by:  tgbtwncom  28466  tgbtwnne  28468  tgbtwnswapid  28470  tgbtwnintr  28471  tgifscgr  28486  tgidinside  28549  tgbtwnconn1lem3  28552  coltr3  28626  mirinv  28644  miriso  28648  krippenlem  28668  midexlem  28670  colperpexlem3  28710  oppne3  28721  oppnid  28724  opphllem1  28725  hlpasch  28734  midid  28759  lmiisolem  28774  f1otrg  28849