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Theorem axtgcont 25168
 Description: Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more information see axtgcont1 25167. (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)
axtgcont.3 (𝜑𝐴𝑃)
axtgcont.4 ((𝜑𝑢𝑆𝑣𝑇) → 𝑢 ∈ (𝐴𝐼𝑣))
Assertion
Ref Expression
axtgcont (𝜑 → ∃𝑏𝑃𝑥𝑆𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦))
Distinct variable groups:   𝑥,𝑦   𝑣,𝑏,𝐴,𝑢,𝑥,𝑦   𝐼,𝑏   𝑣,𝑢,𝑥,𝑦,𝐼   𝑃,𝑏,𝑢,𝑣,𝑥,𝑦   𝑆,𝑏,𝑥   𝑇,𝑏,𝑥,𝑦   ,𝑏,𝑢,𝑣,𝑥,𝑦   𝜑,𝑢,𝑣   𝑢,𝑆,𝑣   𝑢,𝑇,𝑣   𝑢,𝐴,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑏)   𝑆(𝑦)   𝐺(𝑥,𝑦,𝑣,𝑢,𝑏)

Proof of Theorem axtgcont
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 axtgcont.3 . . 3 (𝜑𝐴𝑃)
2 axtgcont.4 . . . . 5 ((𝜑𝑢𝑆𝑣𝑇) → 𝑢 ∈ (𝐴𝐼𝑣))
323expb 1258 . . . 4 ((𝜑 ∧ (𝑢𝑆𝑣𝑇)) → 𝑢 ∈ (𝐴𝐼𝑣))
43ralrimivva 2954 . . 3 (𝜑 → ∀𝑢𝑆𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣))
5 simplr 788 . . . . . . 7 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢)
6 simpll 786 . . . . . . . 8 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑎 = 𝐴)
7 simpr 476 . . . . . . . 8 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
86, 7oveq12d 6567 . . . . . . 7 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑎𝐼𝑦) = (𝐴𝐼𝑣))
95, 8eleq12d 2682 . . . . . 6 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑥 ∈ (𝑎𝐼𝑦) ↔ 𝑢 ∈ (𝐴𝐼𝑣)))
109cbvraldva 3153 . . . . 5 ((𝑎 = 𝐴𝑥 = 𝑢) → (∀𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣)))
1110cbvraldva 3153 . . . 4 (𝑎 = 𝐴 → (∀𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑢𝑆𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣)))
1211rspcev 3282 . . 3 ((𝐴𝑃 ∧ ∀𝑢𝑆𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣)) → ∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦))
131, 4, 12syl2anc 691 . 2 (𝜑 → ∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦))
14 axtrkg.p . . 3 𝑃 = (Base‘𝐺)
15 axtrkg.d . . 3 = (dist‘𝐺)
16 axtrkg.i . . 3 𝐼 = (Itv‘𝐺)
17 axtrkg.g . . 3 (𝜑𝐺 ∈ TarskiG)
18 axtgcont.1 . . 3 (𝜑𝑆𝑃)
19 axtgcont.2 . . 3 (𝜑𝑇𝑃)
2014, 15, 16, 17, 18, 19axtgcont1 25167 . 2 (𝜑 → (∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑆𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦)))
2113, 20mpd 15 1 (𝜑 → ∃𝑏𝑃𝑥𝑆𝑦𝑇 𝑏 ∈ (𝑥𝐼𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135 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:  f1otrg  25551
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