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Theorem tglnssp 26643
Description: Lines are subset of the geometry base set. That is, lines are sets of points. (Contributed by Thierry Arnoux, 17-May-2019.)
Hypotheses
Ref Expression
tglngval.p 𝑃 = (Base‘𝐺)
tglngval.l 𝐿 = (LineG‘𝐺)
tglngval.i 𝐼 = (Itv‘𝐺)
tglngval.g (𝜑𝐺 ∈ TarskiG)
tglngval.x (𝜑𝑋𝑃)
tglngval.y (𝜑𝑌𝑃)
tglngval.z (𝜑𝑋𝑌)
Assertion
Ref Expression
tglnssp (𝜑 → (𝑋𝐿𝑌) ⊆ 𝑃)

Proof of Theorem tglnssp
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 tglngval.p . . 3 𝑃 = (Base‘𝐺)
2 tglngval.l . . 3 𝐿 = (LineG‘𝐺)
3 tglngval.i . . 3 𝐼 = (Itv‘𝐺)
4 tglngval.g . . 3 (𝜑𝐺 ∈ TarskiG)
5 tglngval.x . . 3 (𝜑𝑋𝑃)
6 tglngval.y . . 3 (𝜑𝑌𝑃)
7 tglngval.z . . 3 (𝜑𝑋𝑌)
81, 2, 3, 4, 5, 6, 7tglngval 26642 . 2 (𝜑 → (𝑋𝐿𝑌) = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
9 ssrab2 3993 . 2 {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ⊆ 𝑃
108, 9eqsstrdi 3955 1 (𝜑 → (𝑋𝐿𝑌) ⊆ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1088   = wceq 1543  wcel 2110  wne 2940  {crab 3065  wss 3866  cfv 6380  (class class class)co 7213  Basecbs 16760  TarskiGcstrkg 26521  Itvcitv 26527  LineGclng 26528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-trkg 26544
This theorem is referenced by:  tglineelsb2  26723  tglinecom  26726
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