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Theorem axtgcont1 25167
Description: Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. This axiom (scheme) asserts that any two sets 𝑆 and 𝑇 (of points) such that the elements of 𝑆 precede the elements of 𝑇 with respect to some point 𝑎 (that is, 𝑥 is between 𝑎 and 𝑦 whenever 𝑥 is in 𝑋 and 𝑦 is in 𝑌) are separated by some point 𝑏; this is explained in Axiom 11 of [Tarski1999] p. 185. (Contributed by Thierry Arnoux, 16-Mar-2019.)
Hypotheses
Ref Expression
axtrkg.p 𝑃 = (Base‘𝐺)
axtrkg.d = (dist‘𝐺)
axtrkg.i 𝐼 = (Itv‘𝐺)
axtrkg.g (𝜑𝐺 ∈ TarskiG)
axtgcont.1 (𝜑𝑆𝑃)
axtgcont.2 (𝜑𝑇𝑃)
Assertion
Ref Expression
axtgcont1 (𝜑 → (∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑆𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦)))
Distinct variable groups:   𝑥,𝑦   𝑎,𝑏,𝑥,𝑦,𝐼   𝑃,𝑎,𝑏,𝑥,𝑦   𝑆,𝑎,𝑏,𝑥   𝑇,𝑎,𝑏,𝑥,𝑦   ,𝑎,𝑏,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑎,𝑏)   𝑆(𝑦)   𝐺(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem axtgcont1
Dummy variables 𝑓 𝑖 𝑝 𝑧 𝑣 𝑠 𝑡 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-trkg 25152 . . . . 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 3795 . . . . . 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 3796 . . . . . 6 (TarskiGC ∩ TarskiGB) ⊆ TarskiGB
42, 3sstri 3577 . . . . 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 3598 . . . 4 TarskiG ⊆ TarskiGB
6 axtrkg.g . . . 4 (𝜑𝐺 ∈ TarskiG)
75, 6sseldi 3566 . . 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 25154 . . . . 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 479 . . . 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))))
1312simp3d 1068 . . 3 (𝐺 ∈ TarskiGB → ∀𝑠 ∈ 𝒫 𝑃𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)))
147, 13syl 17 . 2 (𝜑 → ∀𝑠 ∈ 𝒫 𝑃𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)))
15 axtgcont.1 . . . 4 (𝜑𝑆𝑃)
16 fvex 6113 . . . . . . 7 (Base‘𝐺) ∈ V
178, 16eqeltri 2684 . . . . . 6 𝑃 ∈ V
1817ssex 4730 . . . . 5 (𝑆𝑃𝑆 ∈ V)
19 elpwg 4116 . . . . 5 (𝑆 ∈ V → (𝑆 ∈ 𝒫 𝑃𝑆𝑃))
2015, 18, 193syl 18 . . . 4 (𝜑 → (𝑆 ∈ 𝒫 𝑃𝑆𝑃))
2115, 20mpbird 246 . . 3 (𝜑𝑆 ∈ 𝒫 𝑃)
22 axtgcont.2 . . . 4 (𝜑𝑇𝑃)
2317ssex 4730 . . . . 5 (𝑇𝑃𝑇 ∈ V)
24 elpwg 4116 . . . . 5 (𝑇 ∈ V → (𝑇 ∈ 𝒫 𝑃𝑇𝑃))
2522, 23, 243syl 18 . . . 4 (𝜑 → (𝑇 ∈ 𝒫 𝑃𝑇𝑃))
2622, 25mpbird 246 . . 3 (𝜑𝑇 ∈ 𝒫 𝑃)
27 raleq 3115 . . . . . 6 (𝑠 = 𝑆 → (∀𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑥𝑆𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦)))
2827rexbidv 3034 . . . . 5 (𝑠 = 𝑆 → (∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∃𝑎𝑃𝑥𝑆𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦)))
29 raleq 3115 . . . . . 6 (𝑠 = 𝑆 → (∀𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦) ↔ ∀𝑥𝑆𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)))
3029rexbidv 3034 . . . . 5 (𝑠 = 𝑆 → (∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦) ↔ ∃𝑏𝑃𝑥𝑆𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)))
3128, 30imbi12d 333 . . . 4 (𝑠 = 𝑆 → ((∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)) ↔ (∃𝑎𝑃𝑥𝑆𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑆𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦))))
32 raleq 3115 . . . . . 6 (𝑡 = 𝑇 → (∀𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦)))
3332rexralbidv 3040 . . . . 5 (𝑡 = 𝑇 → (∃𝑎𝑃𝑥𝑆𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦)))
34 raleq 3115 . . . . . 6 (𝑡 = 𝑇 → (∀𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦) ↔ ∀𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦)))
3534rexralbidv 3040 . . . . 5 (𝑡 = 𝑇 → (∃𝑏𝑃𝑥𝑆𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦) ↔ ∃𝑏𝑃𝑥𝑆𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦)))
3633, 35imbi12d 333 . . . 4 (𝑡 = 𝑇 → ((∃𝑎𝑃𝑥𝑆𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑆𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)) ↔ (∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑆𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦))))
3731, 36rspc2v 3293 . . 3 ((𝑆 ∈ 𝒫 𝑃𝑇 ∈ 𝒫 𝑃) → (∀𝑠 ∈ 𝒫 𝑃𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)) → (∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑆𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦))))
3821, 26, 37syl2anc 691 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝑃𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)) → (∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑆𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦))))
3914, 38mpd 15 1 (𝜑 → (∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑆𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3o 1030  w3a 1031   = wceq 1475  wcel 1977  {cab 2596  wral 2896  wrex 2897  {crab 2900  Vcvv 3173  [wsbc 3402  cdif 3537  cin 3539  wss 3540  𝒫 cpw 4108  {csn 4125  cfv 5804  (class class class)co 6549  cmpt2 6551  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  TarskiGCcstrkgc 25130  TarskiGBcstrkgb 25131  TarskiGCBcstrkgcb 25132  Itvcitv 25135  LineGclng 25136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-trkgb 25148  df-trkg 25152
This theorem is referenced by:  axtgcont  25168
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