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Theorem axtgcont 26182
Description: Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more information see axtgcont1 26181. (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 1112 . . . 4 ((𝜑 ∧ (𝑢𝑆𝑣𝑇)) → 𝑢 ∈ (𝐴𝐼𝑣))
43ralrimivva 3188 . . 3 (𝜑 → ∀𝑢𝑆𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣))
5 simplr 765 . . . . . . 7 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢)
6 simpll 763 . . . . . . . 8 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑎 = 𝐴)
7 simpr 485 . . . . . . . 8 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
86, 7oveq12d 7163 . . . . . . 7 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑎𝐼𝑦) = (𝐴𝐼𝑣))
95, 8eleq12d 2904 . . . . . 6 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑥 ∈ (𝑎𝐼𝑦) ↔ 𝑢 ∈ (𝐴𝐼𝑣)))
109cbvraldva 3457 . . . . 5 ((𝑎 = 𝐴𝑥 = 𝑢) → (∀𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣)))
1110cbvraldva 3457 . . . 4 (𝑎 = 𝐴 → (∀𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑢𝑆𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣)))
1211rspcev 3620 . . 3 ((𝐴𝑃 ∧ ∀𝑢𝑆𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣)) → ∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦))
131, 4, 12syl2anc 584 . 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 26181 . 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 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  wss 3933  cfv 6348  (class class class)co 7145  Basecbs 16471  distcds 16562  TarskiGcstrkg 26143  Itvcitv 26149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-trkgb 26162  df-trkg 26166
This theorem is referenced by:  f1otrg  26584
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