Theorem tglnpt 24673

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 24672	. . 3 |- ( G e. TarskiG -> U. ran L C_ P )
6	1, 5	syl 17	. 2 |- ( ph -> U. ran L C_ P )
7		tglnpt.a	. . . 4 |- ( ph -> A e. ran L )
8		elssuni 4219	. . . 4 |- ( A e. ran L -> A C_ U. ran L )
9	7, 8	syl 17	. . 3 |- ( ph -> A C_ U. ran L )
10		tglnpt.x	. . 3 |- ( ph -> X e. A )
11	9, 10	sseldd 3419	. 2 |- ( ph -> X e. U. ran L )
12	6, 11	sseldd 3419	1 |- ( ph -> X e. P )

Syntax hints:   -> wi 4   = wceq 1452   e. wcel 1904   C_ wss 3390   U.cuni 4190   ran crn 4840   ` cfv 5589   Basecbs 15199  TarskiGcstrkg 24557  Itvcitv 24563  LineGclng 24564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-cnv 4847  df-dm 4849  df-rn 4850  df-iota 5553  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-trkg 24580

This theorem is referenced by:  mirln  24800  mirln2  24801  perpcom  24837  perpneq  24838  ragperp  24841  foot  24843  footne  24844  footeq  24845  hlperpnel  24846  perprag  24847  perpdragALT  24848  perpdrag  24849  colperpexlem3  24853  oppne3  24864  oppcom  24865  oppnid  24867  opphllem1  24868  opphllem2  24869  opphllem3  24870  opphllem4  24871  opphllem5  24872  opphllem6  24873  oppperpex  24874  opphl  24875  outpasch  24876  lnopp2hpgb  24884  hpgerlem  24886  colopp  24890  colhp  24891  lmieu  24905  lmimid  24915  lnperpex  24924  trgcopy  24925  trgcopyeulem  24926