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Theorem axtgcont 28388
Description: Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more information see axtgcont1 28387. (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 1117 . . . 4 ((𝜑 ∧ (𝑢𝑆𝑣𝑇)) → 𝑢 ∈ (𝐴𝐼𝑣))
43ralrimivva 3190 . . 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 483 . . . . . . . 8 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
86, 7oveq12d 7441 . . . . . . 7 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑎𝐼𝑦) = (𝐴𝐼𝑣))
95, 8eleq12d 2819 . . . . . 6 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑥 ∈ (𝑎𝐼𝑦) ↔ 𝑢 ∈ (𝐴𝐼𝑣)))
109cbvraldva 3226 . . . . 5 ((𝑎 = 𝐴𝑥 = 𝑢) → (∀𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣)))
1110cbvraldva 3226 . . . 4 (𝑎 = 𝐴 → (∀𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑢𝑆𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣)))
1211rspcev 3607 . . 3 ((𝐴𝑃 ∧ ∀𝑢𝑆𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣)) → ∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦))
131, 4, 12syl2anc 582 . 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 28387 . 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 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  wrex 3059  wss 3946  cfv 6553  (class class class)co 7423  Basecbs 17208  distcds 17270  TarskiGcstrkg 28346  Itvcitv 28352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5303  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7426  df-trkgb 28368  df-trkg 28372
This theorem is referenced by:  f1otrg  28790
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