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Theorem tglnssp 26040
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 26039 . 2 (𝜑 → (𝑋𝐿𝑌) = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
9 ssrab2 3947 . 2 {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ⊆ 𝑃
108, 9syl6eqss 3912 1 (𝜑 → (𝑋𝐿𝑌) ⊆ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1067   = wceq 1507  wcel 2050  wne 2968  {crab 3093  wss 3830  cfv 6188  (class class class)co 6976  Basecbs 16339  TarskiGcstrkg 25918  Itvcitv 25924  LineGclng 25925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-iota 6152  df-fun 6190  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-trkg 25941
This theorem is referenced by:  tglineelsb2  26120  tglinecom  26123
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