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Theorem axtglowdim2ALTV 34235
Description: Alternate version of axtglowdim2 28261. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.)
Hypotheses
Ref Expression
istrkg2d.p 𝑃 = (Baseβ€˜πΊ)
istrkg2d.d βˆ’ = (distβ€˜πΊ)
istrkg2d.i 𝐼 = (Itvβ€˜πΊ)
axtglowdim2ALTV.g (πœ‘ β†’ 𝐺 ∈ TarskiG2D)
Assertion
Ref Expression
axtglowdim2ALTV (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘§ ∈ 𝑃 Β¬ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧)))
Distinct variable groups:   π‘₯, βˆ’ ,𝑦,𝑧   π‘₯,𝐼,𝑦,𝑧   π‘₯,𝑃,𝑦,𝑧
Allowed substitution hints:   πœ‘(π‘₯,𝑦,𝑧)   𝐺(π‘₯,𝑦,𝑧)

Proof of Theorem axtglowdim2ALTV
Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axtglowdim2ALTV.g . . . 4 (πœ‘ β†’ 𝐺 ∈ TarskiG2D)
2 istrkg2d.p . . . . 5 𝑃 = (Baseβ€˜πΊ)
3 istrkg2d.d . . . . 5 βˆ’ = (distβ€˜πΊ)
4 istrkg2d.i . . . . 5 𝐼 = (Itvβ€˜πΊ)
52, 3, 4istrkg2d 34234 . . . 4 (𝐺 ∈ TarskiG2D ↔ (𝐺 ∈ V ∧ (βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘§ ∈ 𝑃 Β¬ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧)) ∧ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((π‘₯ βˆ’ 𝑒) = (π‘₯ βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))))))
61, 5sylib 217 . . 3 (πœ‘ β†’ (𝐺 ∈ V ∧ (βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘§ ∈ 𝑃 Β¬ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧)) ∧ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((π‘₯ βˆ’ 𝑒) = (π‘₯ βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))))))
76simprd 495 . 2 (πœ‘ β†’ (βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘§ ∈ 𝑃 Β¬ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧)) ∧ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((π‘₯ βˆ’ 𝑒) = (π‘₯ βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧)))))
87simpld 494 1 (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘§ ∈ 𝑃 Β¬ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∨ w3o 1084   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  βˆ€wral 3056  βˆƒwrex 3065  Vcvv 3469  β€˜cfv 6542  (class class class)co 7414  Basecbs 17171  distcds 17233  Itvcitv 28224  TarskiG2Dcstrkg2d 34232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-nul 5300
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417  df-trkg2d 34233
This theorem is referenced by: (None)
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