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Theorem tglnne0 28431
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 2727 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2727 . . . . 5 (Itv‘𝐺) = (Itv‘𝐺)
3 tglnne0.l . . . . 5 𝐿 = (LineG‘𝐺)
4 tglnne0.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 729 . . . . 5 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝐺 ∈ TarskiG)
6 simpllr 775 . . . . 5 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (Base‘𝐺))
7 simplr 768 . . . . 5 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑦 ∈ (Base‘𝐺))
8 simprr 772 . . . . 5 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥𝑦)
91, 2, 3, 5, 6, 7, 8tglinerflx1 28424 . . . 4 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦))
10 simprl 770 . . . 4 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝐴 = (𝑥𝐿𝑦))
119, 10eleqtrrd 2831 . . 3 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥𝐴)
1211ne0d 4331 . 2 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝐴 ≠ ∅)
13 tglnne0.1 . . 3 (𝜑𝐴 ∈ ran 𝐿)
141, 2, 3, 4, 13tgisline 28418 . 2 (𝜑 → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦))
1512, 14r19.29vva 3208 1 (𝜑𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wne 2935  c0 4318  ran crn 5673  cfv 6542  (class class class)co 7414  Basecbs 17171  TarskiGcstrkg 28218  Itvcitv 28224  LineGclng 28225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-trkgc 28239  df-trkgb 28240  df-trkgcb 28241  df-trkg 28244
This theorem is referenced by:  hpgerlem  28556
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