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Theorem axtgcont 26254
Description: Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more information see axtgcont1 26253. (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 1116 . . . 4 ((𝜑 ∧ (𝑢𝑆𝑣𝑇)) → 𝑢 ∈ (𝐴𝐼𝑣))
43ralrimivva 3191 . . 3 (𝜑 → ∀𝑢𝑆𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣))
5 simplr 767 . . . . . . 7 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢)
6 simpll 765 . . . . . . . 8 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑎 = 𝐴)
7 simpr 487 . . . . . . . 8 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
86, 7oveq12d 7173 . . . . . . 7 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑎𝐼𝑦) = (𝐴𝐼𝑣))
95, 8eleq12d 2907 . . . . . 6 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑥 ∈ (𝑎𝐼𝑦) ↔ 𝑢 ∈ (𝐴𝐼𝑣)))
109cbvraldva 3459 . . . . 5 ((𝑎 = 𝐴𝑥 = 𝑢) → (∀𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣)))
1110cbvraldva 3459 . . . 4 (𝑎 = 𝐴 → (∀𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑢𝑆𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣)))
1211rspcev 3622 . . 3 ((𝐴𝑃 ∧ ∀𝑢𝑆𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣)) → ∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦))
131, 4, 12syl2anc 586 . 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 26253 . 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 398  w3a 1083   = wceq 1533  wcel 2110  wral 3138  wrex 3139  wss 3935  cfv 6354  (class class class)co 7155  Basecbs 16482  distcds 16573  TarskiGcstrkg 26215  Itvcitv 26221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-iota 6313  df-fv 6362  df-ov 7158  df-trkgb 26234  df-trkg 26238
This theorem is referenced by:  f1otrg  26656
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