Theorem tglnssp 29024

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglngval.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
tglngval.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
tglngval.z	⊢ (𝜑 → 𝑋 ≠ 𝑌)

Assertion

Ref	Expression
tglnssp	⊢ (𝜑 → (𝑋𝐿𝑌) ⊆ 𝑃)

Proof of Theorem tglnssp

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	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 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		tglngval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
6		tglngval.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
7		tglngval.z	. . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
8	1, 2, 3, 4, 5, 6, 7	tglngval 29023	. 2 ⊢ (𝜑 → (𝑋𝐿𝑌) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
9		ssrab2 4004	. 2 ⊢ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ⊆ 𝑃
10	8, 9	eqsstrdi 3953	1 ⊢ (𝜑 → (𝑋𝐿𝑌) ⊆ 𝑃)

This proof depends on syntax axioms:   → wi 4   ∨ w3o 1102   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  {crab 3412   ⊆ wss 3896  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  TarskiGcstrkg 28898  Itvcitv 28904  LineGclng 28905

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6497  df-fun 6545  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-trkg 28924

This theorem is used by:  tglineelsb2  29109  tglinecom  29112