Theorem tglnpt 28799

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 28798	. . 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 4905	. . . 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 3939	. 2 ⊢ (𝜑 → 𝑋 ∈ ∪ ran 𝐿)
12	6, 11	sseldd 3939	1 ⊢ (𝜑 → 𝑋 ∈ 𝑃)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143   ⊆ wss 3906  ∪ cuni 4873  ran crn 5664  ‘cfv 6538  Basecbs 17270  TarskiGcstrkg 28677  Itvcitv 28683  LineGclng 28684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-cnv 5671  df-dm 5673  df-rn 5674  df-iota 6494  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-trkg 28703

This theorem is referenced by:  tglnpt3  28908  mirln  28934  mirln2  28935  symquadprlnglem  28951  perpcom  28974  perpneq  28975  ragperp  28978  foot  28983  footne  28984  footeq  28985  hlperpnel  28987  perprag  28988  perpdragALT  28989  perpdrag  28990  colperpexlem3  28994  oppne3  29005  oppcom  29006  oppnid  29008  opphllem1  29009  opphllem2  29010  opphllem3  29011  opphllem4  29012  opphllem5  29013  opphllem6  29014  oppperpex  29015  opphl  29016  oppmir  29017  outpasch  29018  lnopp2hpgb  29026  hpgerlem  29028  colopp  29032  colhp  29033  hlopp  29035  elplnglnid  29046  lnincplng  29047  plngrotlem1  29050  lnssplnglem  29054  plngmiropp  29057  nhpmirhp  29061  lmieu  29074  lmimid  29084  symquadmid  29089  lnperpex  29094  trgcopy  29096  trgcopyeulem  29097  perpeqlem  29131  perpeq  29132  prlnghpg  29177  prlngpln3  29180  perpprlng  29181  prlngex  29182  prlngmolem1  29183  prlngmolem2  29184  prlngeq  29188  prlngplngtr  29190  prlngmid2  29192  symquadprlng  29193  quadcgrprlng  29197