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Theorem tglnssp 25247
 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 25246 . 2 (𝜑 → (𝑋𝐿𝑌) = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
9 ssrab2 3650 . 2 {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ⊆ 𝑃
108, 9syl6eqss 3618 1 (𝜑 → (𝑋𝐿𝑌) ⊆ 𝑃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  TarskiGcstrkg 25129  Itvcitv 25135  LineGclng 25136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-trkg 25152 This theorem is referenced by:  tglineelsb2  25327  tglinecom  25330
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