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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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