Theorem tglnpt 25860

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 25859	. . 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 4688	. . . 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 3827	. 2 ⊢ (𝜑 → 𝑋 ∈ ∪ ran 𝐿)
12	6, 11	sseldd 3827	1 ⊢ (𝜑 → 𝑋 ∈ 𝑃)

Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166   ⊆ wss 3797  ∪ cuni 4657  ran crn 5342  ‘cfv 6122  Basecbs 16221  TarskiGcstrkg 25741  Itvcitv 25747  LineGclng 25748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-cnv 5349  df-dm 5351  df-rn 5352  df-iota 6085  df-fv 6130  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-trkg 25764

This theorem is referenced by:  mirln  25987  mirln2  25988  perpcom  26024  perpneq  26025  ragperp  26028  foot  26030  footne  26031  footeq  26032  hlperpnel  26033  perprag  26034  perpdragALT  26035  perpdrag  26036  colperpexlem3  26040  oppne3  26051  oppcom  26052  oppnid  26054  opphllem1  26055  opphllem2  26056  opphllem3  26057  opphllem4  26058  opphllem5  26059  opphllem6  26060  oppperpex  26061  opphl  26062  outpasch  26063  lnopp2hpgb  26071  hpgerlem  26073  colopp  26077  colhp  26078  lmieu  26092  lmimid  26102  lnperpex  26111  trgcopy  26112  trgcopyeulem  26113