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Theorem axtgbtwnid 26179
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 26166 . . . . 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 4202 . . . . . 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 4203 . . . . . 6 (TarskiGC ∩ TarskiGB) ⊆ TarskiGB
42, 3sstri 3973 . . . . 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 3998 . . . 4 TarskiG ⊆ TarskiGB
6 axtrkg.g . . . 4 (𝜑𝐺 ∈ TarskiG)
75, 6sseldi 3962 . . 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 26168 . . . . 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 497 . . . 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 1134 . . 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 7163 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐼𝑥) = (𝑋𝐼𝑋))
2019eleq2d 2895 . . . . 5 (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑥) ↔ 𝑦 ∈ (𝑋𝐼𝑋)))
21 eqeq1 2822 . . . . 5 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
2220, 21imbi12d 346 . . . 4 (𝑥 = 𝑋 → ((𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ↔ (𝑦 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑦)))
23 eleq1 2897 . . . . 5 (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑋) ↔ 𝑌 ∈ (𝑋𝐼𝑋)))
24 eqeq2 2830 . . . . 5 (𝑦 = 𝑌 → (𝑋 = 𝑦𝑋 = 𝑌))
2523, 24imbi12d 346 . . . 4 (𝑦 = 𝑌 → ((𝑦 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑦) ↔ (𝑌 ∈ (𝑋𝐼𝑋) → 𝑋 = 𝑌)))
2622, 25rspc2v 3630 . . 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 396  w3o 1078  w3a 1079   = wceq 1528  wcel 2105  {cab 2796  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  [wsbc 3769  cdif 3930  cin 3932  𝒫 cpw 4535  {csn 4557  cfv 6348  (class class class)co 7145  cmpo 7147  Basecbs 16471  distcds 16562  TarskiGcstrkg 26143  TarskiGCcstrkgc 26144  TarskiGBcstrkgb 26145  TarskiGCBcstrkgcb 26146  Itvcitv 26149  LineGclng 26150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-trkgb 26162  df-trkg 26166
This theorem is referenced by:  tgbtwncom  26201  tgbtwnne  26203  tgbtwnswapid  26205  tgbtwnintr  26206  tgifscgr  26221  tgidinside  26284  tgbtwnconn1lem3  26287  coltr3  26361  mirinv  26379  miriso  26383  krippenlem  26403  midexlem  26405  colperpexlem3  26445  oppne3  26456  oppnid  26459  opphllem1  26460  hlpasch  26469  midid  26494  lmiisolem  26509  f1otrg  26584
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