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Theorem axtgcont 25781
Description: Axiom of Continuity. Axiom A11 of [Schwabhauser] p. 13. For more information see axtgcont1 25780. (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 1155 . . . 4 ((𝜑 ∧ (𝑢𝑆𝑣𝑇)) → 𝑢 ∈ (𝐴𝐼𝑣))
43ralrimivva 3180 . . 3 (𝜑 → ∀𝑢𝑆𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣))
5 simplr 787 . . . . . . 7 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢)
6 simpll 785 . . . . . . . 8 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑎 = 𝐴)
7 simpr 479 . . . . . . . 8 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
86, 7oveq12d 6923 . . . . . . 7 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑎𝐼𝑦) = (𝐴𝐼𝑣))
95, 8eleq12d 2900 . . . . . 6 (((𝑎 = 𝐴𝑥 = 𝑢) ∧ 𝑦 = 𝑣) → (𝑥 ∈ (𝑎𝐼𝑦) ↔ 𝑢 ∈ (𝐴𝐼𝑣)))
109cbvraldva 3389 . . . . 5 ((𝑎 = 𝐴𝑥 = 𝑢) → (∀𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣)))
1110cbvraldva 3389 . . . 4 (𝑎 = 𝐴 → (∀𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦) ↔ ∀𝑢𝑆𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣)))
1211rspcev 3526 . . 3 ((𝐴𝑃 ∧ ∀𝑢𝑆𝑣𝑇 𝑢 ∈ (𝐴𝐼𝑣)) → ∃𝑎𝑃𝑥𝑆𝑦𝑇 𝑥 ∈ (𝑎𝐼𝑦))
131, 4, 12syl2anc 581 . 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 25780 . 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 386  w3a 1113   = wceq 1658  wcel 2166  wral 3117  wrex 3118  wss 3798  cfv 6123  (class class class)co 6905  Basecbs 16222  distcds 16314  TarskiGcstrkg 25742  Itvcitv 25748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908  df-trkgb 25761  df-trkg 25765
This theorem is referenced by:  f1otrg  26170
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