Theorem tglnpt 24137

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	|- ( ph -> G e. TarskiG )
tglnpt.a	|- ( ph -> A e. ran L )
tglnpt.x	|- ( ph -> X e. A )

Assertion

Ref	Expression
tglnpt	|- ( ph -> X e. P )

Proof of Theorem tglnpt

Step	Hyp	Ref	Expression
1		tglnpt.g	. . 3 |- ( ph -> G e. 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 24136	. . 3 |- ( G e. TarskiG -> U. ran L C_ P )
6	1, 5	syl 16	. 2 |- ( ph -> U. ran L C_ P )
7		tglnpt.a	. . . 4 |- ( ph -> A e. ran L )
8		elssuni 4264	. . . 4 |- ( A e. ran L -> A C_ U. ran L )
9	7, 8	syl 16	. . 3 |- ( ph -> A C_ U. ran L )
10		tglnpt.x	. . 3 |- ( ph -> X e. A )
11	9, 10	sseldd 3490	. 2 |- ( ph -> X e. U. ran L )
12	6, 11	sseldd 3490	1 |- ( ph -> X e. P )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 1823   C_ wss 3461   U.cuni 4235   ran crn 4989   ` cfv 5570   Basecbs 14716  TarskiGcstrkg 24023  Itvcitv 24030  LineGclng 24031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-cnv 4996  df-dm 4998  df-rn 4999  df-iota 5534  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-trkg 24048

This theorem is referenced by:  mirln  24257  mirln2  24258  perpcom  24291  perpneq  24292  ragperp  24295  foot  24297  footne  24298  footeq  24299  hlperpnel  24300  perprag  24301  perpdragALT  24302  perpdrag  24303  colperpexlem3  24307  oppcom  24317  oppnid  24319  opphllem1  24320  opphllem2  24321  opphllem3  24322  opphllem4  24323  opphllem5  24324  opphllem6  24325  opphl  24326  lnopp2hpgb  24333  hpgerlem  24335  lmieu  24351  lmimid  24360