MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tglnne0 Structured version   Visualization version   GIF version

Theorem tglnne0 25952
Description: A line 𝐴 has at least one point. (Contributed by Thierry Arnoux, 4-Mar-2020.)
Hypotheses
Ref Expression
tglnne0.l 𝐿 = (LineG‘𝐺)
tglnne0.g (𝜑𝐺 ∈ TarskiG)
tglnne0.1 (𝜑𝐴 ∈ ran 𝐿)
Assertion
Ref Expression
tglnne0 (𝜑𝐴 ≠ ∅)

Proof of Theorem tglnne0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2825 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2825 . . . . 5 (Itv‘𝐺) = (Itv‘𝐺)
3 tglnne0.l . . . . 5 𝐿 = (LineG‘𝐺)
4 tglnne0.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 721 . . . . 5 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝐺 ∈ TarskiG)
6 simpllr 793 . . . . 5 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (Base‘𝐺))
7 simplr 785 . . . . 5 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑦 ∈ (Base‘𝐺))
8 simprr 789 . . . . 5 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥𝑦)
91, 2, 3, 5, 6, 7, 8tglinerflx1 25945 . . . 4 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦))
10 simprl 787 . . . 4 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝐴 = (𝑥𝐿𝑦))
119, 10eleqtrrd 2909 . . 3 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥𝐴)
1211ne0d 4151 . 2 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝐴 ≠ ∅)
13 tglnne0.1 . . 3 (𝜑𝐴 ∈ ran 𝐿)
141, 2, 3, 4, 13tgisline 25939 . 2 (𝜑 → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦))
1512, 14r19.29vva 3291 1 (𝜑𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  wne 2999  c0 4144  ran crn 5343  cfv 6123  (class class class)co 6905  Basecbs 16222  TarskiGcstrkg 25742  Itvcitv 25748  LineGclng 25749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-trkgc 25760  df-trkgb 25761  df-trkgcb 25762  df-trkg 25765
This theorem is referenced by:  hpgerlem  26074
  Copyright terms: Public domain W3C validator