Theorem tglnpt 28855

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 28854	. . 3 ⊢ (𝐺 ∈ TarskiG → ∪ ran 𝐿 ⊆ 𝑃)
6	1, 5	syl 18	. 2 ⊢ (𝜑 → ∪ ran 𝐿 ⊆ 𝑃)
7		tglnpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
8		elssuni 4909	. . . 4 ⊢ (𝐴 ∈ ran 𝐿 → 𝐴 ⊆ ∪ ran 𝐿)
9	7, 8	syl 18	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran 𝐿)
10		tglnpt.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
11	9, 10	sseldd 3941	. 2 ⊢ (𝜑 → 𝑋 ∈ ∪ ran 𝐿)
12	6, 11	sseldd 3941	1 ⊢ (𝜑 → 𝑋 ∈ 𝑃)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146   ⊆ wss 3908  ∪ cuni 4877  ran crn 5667  ‘cfv 6543  Basecbs 17294  TarskiGcstrkg 28733  Itvcitv 28739  LineGclng 28740

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-cnv 5674  df-dm 5676  df-rn 5677  df-iota 6499  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-trkg 28759

This theorem is used by:  tglnpt3  28964  mirln  28990  mirln2  28991  symquadprlnglem  29007  perpcom  29030  perpneq  29031  ragperp  29034  foot  29039  footne  29040  footeq  29041  hlperpnel  29043  perprag  29044  perpdragALT  29045  perpdrag  29046  colperpexlem3  29050  oppne3  29061  oppcom  29062  oppnid  29064  opphllem1  29065  opphllem2  29066  opphllem3  29067  opphllem4  29068  opphllem5  29069  opphllem6  29070  oppperpex  29071  opphl  29072  oppmir  29073  outpasch  29074  lnopp2hpgb  29082  hpgerlem  29084  colopp  29088  colhp  29089  hlopp  29091  elplnglnid  29102  lnincplng  29103  plngrotlem1  29106  lnssplnglem  29110  plngmiropp  29113  nhpmirhp  29117  lmieu  29130  lmimid  29140  symquadmid  29145  lnperpex  29150  trgcopy  29152  trgcopyeulem  29153  perpeqlem  29187  perpeq  29188  prlnghpg  29233  prlngpln3  29236  perpprlng  29237  prlngex  29238  prlngmolem1  29239  prlngmolem2  29240  prlngeq  29244  prlngplngtr  29246  prlngmid2  29248  symquadprlng  29249  quadcgrprlng  29253