Theorem tglnpt 28572

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 28571	. . 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 4942	. . . 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 3996	. 2 ⊢ (𝜑 → 𝑋 ∈ ∪ ran 𝐿)
12	6, 11	sseldd 3996	1 ⊢ (𝜑 → 𝑋 ∈ 𝑃)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106   ⊆ wss 3963  ∪ cuni 4912  ran crn 5690  ‘cfv 6563  Basecbs 17245  TarskiGcstrkg 28450  Itvcitv 28456  LineGclng 28457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-iota 6516  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-trkg 28476

This theorem is referenced by:  mirln  28699  mirln2  28700  perpcom  28736  perpneq  28737  ragperp  28740  foot  28745  footne  28746  footeq  28747  hlperpnel  28748  perprag  28749  perpdragALT  28750  perpdrag  28751  colperpexlem3  28755  oppne3  28766  oppcom  28767  oppnid  28769  opphllem1  28770  opphllem2  28771  opphllem3  28772  opphllem4  28773  opphllem5  28774  opphllem6  28775  oppperpex  28776  opphl  28777  outpasch  28778  lnopp2hpgb  28786  hpgerlem  28788  colopp  28792  colhp  28793  lmieu  28807  lmimid  28817  lnperpex  28826  trgcopy  28827  trgcopyeulem  28828