Theorem tglnpt 28557

Description: Lines are sets of points. (Contributed by Thierry Arnoux, 17-Oct-2019.)

Hypotheses

Ref	Expression
tglng.p	⊢ 𝑃 = (Base‘𝐺)
tglng.l	⊢ 𝐿 = (LineG‘𝐺)
tglng.i	⊢ 𝐼 = (Itv‘𝐺)
tglnpt.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglnpt.a	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
tglnpt.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)

Assertion

Ref	Expression
tglnpt	⊢ (𝜑 → 𝑋 ∈ 𝑃)

Proof of Theorem tglnpt

Step	Hyp	Ref	Expression
1		tglnpt.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
2		tglng.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
3		tglng.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
4		tglng.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
5	2, 3, 4	tglnunirn 28556	. . 3 ⊢ (𝐺 ∈ TarskiG → ∪ ran 𝐿 ⊆ 𝑃)
6	1, 5	syl 17	. 2 ⊢ (𝜑 → ∪ ran 𝐿 ⊆ 𝑃)
7		tglnpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
8		elssuni 4937	. . . 4 ⊢ (𝐴 ∈ ran 𝐿 → 𝐴 ⊆ ∪ ran 𝐿)
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran 𝐿)
10		tglnpt.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
11	9, 10	sseldd 3984	. 2 ⊢ (𝜑 → 𝑋 ∈ ∪ ran 𝐿)
12	6, 11	sseldd 3984	1 ⊢ (𝜑 → 𝑋 ∈ 𝑃)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  ∪ cuni 4907  ran crn 5686  ‘cfv 6561  Basecbs 17247  TarskiGcstrkg 28435  Itvcitv 28441  LineGclng 28442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-iota 6514  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-trkg 28461

This theorem is referenced by:  mirln  28684  mirln2  28685  perpcom  28721  perpneq  28722  ragperp  28725  foot  28730  footne  28731  footeq  28732  hlperpnel  28733  perprag  28734  perpdragALT  28735  perpdrag  28736  colperpexlem3  28740  oppne3  28751  oppcom  28752  oppnid  28754  opphllem1  28755  opphllem2  28756  opphllem3  28757  opphllem4  28758  opphllem5  28759  opphllem6  28760  oppperpex  28761  opphl  28762  outpasch  28763  lnopp2hpgb  28771  hpgerlem  28773  colopp  28777  colhp  28778  lmieu  28792  lmimid  28802  lnperpex  28811  trgcopy  28812  trgcopyeulem  28813