Theorem tglnssp 28307

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 28306	. 2 ⊢ (𝜑 → (𝑋𝐿𝑌) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
9		ssrab2 4072	. 2 ⊢ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ⊆ 𝑃
10	8, 9	eqsstrdi 4031	1 ⊢ (𝜑 → (𝑋𝐿𝑌) ⊆ 𝑃)

Syntax hints:   → wi 4   ∨ w3o 1083   = wceq 1533   ∈ wcel 2098   ≠ wne 2934  {crab 3426   ⊆ wss 3943  ‘cfv 6536  (class class class)co 7404  Basecbs 17151  TarskiGcstrkg 28182  Itvcitv 28188  LineGclng 28189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-trkg 28208

This theorem is referenced by:  tglineelsb2  28387  tglinecom  28390