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Theorem tglnssp 26390
 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 26389 . 2 (𝜑 → (𝑋𝐿𝑌) = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
9 ssrab2 4009 . 2 {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ⊆ 𝑃
108, 9eqsstrdi 3971 1 (𝜑 → (𝑋𝐿𝑌) ⊆ 𝑃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ w3o 1083   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  {crab 3110   ⊆ wss 3883  ‘cfv 6332  (class class class)co 7145  Basecbs 16495  TarskiGcstrkg 26268  Itvcitv 26274  LineGclng 26275 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6291  df-fun 6334  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-trkg 26291 This theorem is referenced by:  tglineelsb2  26470  tglinecom  26473
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