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Theorem axtgcont1 28476
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 28461 . . . . 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 4237 . . . . . 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 4238 . . . . . 6 (TarskiGC ∩ TarskiGB) ⊆ TarskiGB
42, 3sstri 3993 . . . . 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 4030 . . . 4 TarskiG ⊆ TarskiGB
6 axtrkg.g . . . 4 (𝜑𝐺 ∈ TarskiG)
75, 6sselid 3981 . . 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 28463 . . . . 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))))
1312simp3d 1145 . . 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)
168fvexi 6920 . . . . . 6 𝑃 ∈ V
1716ssex 5321 . . . . 5 (𝑆𝑃𝑆 ∈ V)
18 elpwg 4603 . . . . 5 (𝑆 ∈ V → (𝑆 ∈ 𝒫 𝑃𝑆𝑃))
1915, 17, 183syl 18 . . . 4 (𝜑 → (𝑆 ∈ 𝒫 𝑃𝑆𝑃))
2015, 19mpbird 257 . . 3 (𝜑𝑆 ∈ 𝒫 𝑃)
21 axtgcont.2 . . . 4 (𝜑𝑇𝑃)
2216ssex 5321 . . . . 5 (𝑇𝑃𝑇 ∈ V)
23 elpwg 4603 . . . . 5 (𝑇 ∈ V → (𝑇 ∈ 𝒫 𝑃𝑇𝑃))
2421, 22, 233syl 18 . . . 4 (𝜑 → (𝑇 ∈ 𝒫 𝑃𝑇𝑃))
2521, 24mpbird 257 . . 3 (𝜑𝑇 ∈ 𝒫 𝑃)
26 raleq 3323 . . . . . 6 (𝑠 = 𝑆 → (∀𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑥𝑆𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦)))
2726rexbidv 3179 . . . . 5 (𝑠 = 𝑆 → (∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∃𝑎𝑃𝑥𝑆𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦)))
28 raleq 3323 . . . . . 6 (𝑠 = 𝑆 → (∀𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦) ↔ ∀𝑥𝑆𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)))
2928rexbidv 3179 . . . . 5 (𝑠 = 𝑆 → (∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦) ↔ ∃𝑏𝑃𝑥𝑆𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)))
3027, 29imbi12d 344 . . . 4 (𝑠 = 𝑆 → ((∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)) ↔ (∃𝑎𝑃𝑥𝑆𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑆𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦))))
31 raleq 3323 . . . . . 6 (𝑡 = 𝑇 → (∀𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦)))
3231rexralbidv 3223 . . . . 5 (𝑡 = 𝑇 → (∃𝑎𝑃𝑥𝑆𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦)))
33 raleq 3323 . . . . . 6 (𝑡 = 𝑇 → (∀𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦) ↔ ∀𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦)))
3433rexralbidv 3223 . . . . 5 (𝑡 = 𝑇 → (∃𝑏𝑃𝑥𝑆𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦) ↔ ∃𝑏𝑃𝑥𝑆𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦)))
3532, 34imbi12d 344 . . . 4 (𝑡 = 𝑇 → ((∃𝑎𝑃𝑥𝑆𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑆𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)) ↔ (∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑆𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦))))
3630, 35rspc2v 3633 . . 3 ((𝑆 ∈ 𝒫 𝑃𝑇 ∈ 𝒫 𝑃) → (∀𝑠 ∈ 𝒫 𝑃𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)) → (∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑆𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦))))
3720, 25, 36syl2anc 584 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝑃𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)) → (∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑆𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦))))
3814, 37mpd 15 1 (𝜑 → (∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑆𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3o 1086  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  [wsbc 3788  cdif 3948  cin 3950  wss 3951  𝒫 cpw 4600  {csn 4626  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  TarskiGCcstrkgc 28436  TarskiGBcstrkgb 28437  TarskiGCBcstrkgcb 28438  Itvcitv 28441  LineGclng 28442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-trkgb 28457  df-trkg 28461
This theorem is referenced by:  axtgcont  28477
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