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Theorem axtgbtwnid 28488
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 (𝜑𝐺 ∈ TarskiG)
axtgbtwnid.1 (𝜑𝑋𝑃)
axtgbtwnid.2 (𝜑𝑌𝑃)
axtgbtwnid.3 (𝜑𝑌 ∈ (𝑋𝐼𝑋))
Assertion
Ref Expression
axtgbtwnid (𝜑𝑋 = 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.
StepHypRef Expression
1 df-trkg 28475 . . . . 5 TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
2 inss1 4244 . . . . . 6 ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ (TarskiGC ∩ TarskiGB)
3 inss2 4245 . . . . . 6 (TarskiGC ∩ TarskiGB) ⊆ TarskiGB
42, 3sstri 4004 . . . . 5 ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ TarskiGB
51, 4eqsstri 4029 . . . 4 TarskiG ⊆ TarskiGB
6 axtrkg.g . . . 4 (𝜑𝐺 ∈ TarskiG)
75, 6sselid 3992 . . 3 (𝜑𝐺 ∈ TarskiGB)
8 axtrkg.p . . . . . 6 𝑃 = (Base‘𝐺)
9 axtrkg.d . . . . . 6 = (dist‘𝐺)
10 axtrkg.i . . . . . 6 𝐼 = (Itv‘𝐺)
118, 9, 10istrkgb 28477 . . . . 5 (𝐺 ∈ TarskiGB ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)))))
1211simprbi 496 . . . 4 (𝐺 ∈ TarskiGB → (∀𝑥𝑃𝑦𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦))))
1312simp1d 1141 . . 3 (𝐺 ∈ TarskiGB → ∀𝑥𝑃𝑦𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦))
147, 13syl 17 . 2 (𝜑 → ∀𝑥𝑃𝑦𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦))
15 axtgbtwnid.3 . 2 (𝜑𝑌 ∈ (𝑋𝐼𝑋))
16 axtgbtwnid.1 . . 3 (𝜑𝑋𝑃)
17 axtgbtwnid.2 . . 3 (𝜑𝑌𝑃)
18 id 22 . . . . . . 7 (𝑥 = 𝑋𝑥 = 𝑋)
1918, 18oveq12d 7448 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐼𝑥) = (𝑋𝐼𝑋))
2019eleq2d 2824 . . . . 5 (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑥) ↔ 𝑦 ∈ (𝑋𝐼𝑋)))
21 eqeq1 2738 . . . . 5 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
2220, 21imbi12d 344 . . . 4 (𝑥 = 𝑋 → ((𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ↔ (𝑦 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑦)))
23 eleq1 2826 . . . . 5 (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑋) ↔ 𝑌 ∈ (𝑋𝐼𝑋)))
24 eqeq2 2746 . . . . 5 (𝑦 = 𝑌 → (𝑋 = 𝑦𝑋 = 𝑌))
2523, 24imbi12d 344 . . . 4 (𝑦 = 𝑌 → ((𝑦 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑦) ↔ (𝑌 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑌)))
2622, 25rspc2v 3632 . . 3 ((𝑋𝑃𝑌𝑃) → (∀𝑥𝑃𝑦𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) → (𝑌 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑌)))
2716, 17, 26syl2anc 584 . 2 (𝜑 → (∀𝑥𝑃𝑦𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) → (𝑌 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑌)))
2814, 15, 27mp2d 49 1 (𝜑𝑋 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3o 1085  w3a 1086   = wceq 1536  wcel 2105  {cab 2711  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  [wsbc 3790  cdif 3959  cin 3961  𝒫 cpw 4604  {csn 4630  cfv 6562  (class class class)co 7430  cmpo 7432  Basecbs 17244  distcds 17306  TarskiGcstrkg 28449  TarskiGCcstrkgc 28450  TarskiGBcstrkgb 28451  TarskiGCBcstrkgcb 28452  Itvcitv 28455  LineGclng 28456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-nul 5311
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433  df-trkgb 28471  df-trkg 28475
This theorem is referenced by:  tgbtwncom  28510  tgbtwnne  28512  tgbtwnswapid  28514  tgbtwnintr  28515  tgifscgr  28530  tgidinside  28593  tgbtwnconn1lem3  28596  coltr3  28670  mirinv  28688  miriso  28692  krippenlem  28712  midexlem  28714  colperpexlem3  28754  oppne3  28765  oppnid  28768  opphllem1  28769  hlpasch  28778  midid  28803  lmiisolem  28818  f1otrg  28893
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