Theorem tglnssp 28894

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 28893	. 2 ⊢ (𝜑 → (𝑋𝐿𝑌) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
9		ssrab2 4028	. 2 ⊢ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ⊆ 𝑃
10	8, 9	eqsstrdi 3975	1 ⊢ (𝜑 → (𝑋𝐿𝑌) ⊆ 𝑃)

This proof depends on syntax axioms:   → wi 4   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  {crab 3412   ⊆ wss 3899  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  TarskiGcstrkg 28768  Itvcitv 28774  LineGclng 28775

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-trkg 28794

This theorem is used by:  tglineelsb2  28979  tglinecom  28982