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Theorem tglnne0 26420
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 2821 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2821 . . . . 5 (Itv‘𝐺) = (Itv‘𝐺)
3 tglnne0.l . . . . 5 𝐿 = (LineG‘𝐺)
4 tglnne0.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 728 . . . . 5 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝐺 ∈ TarskiG)
6 simpllr 774 . . . . 5 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (Base‘𝐺))
7 simplr 767 . . . . 5 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑦 ∈ (Base‘𝐺))
8 simprr 771 . . . . 5 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥𝑦)
91, 2, 3, 5, 6, 7, 8tglinerflx1 26413 . . . 4 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦))
10 simprl 769 . . . 4 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝐴 = (𝑥𝐿𝑦))
119, 10eleqtrrd 2916 . . 3 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝑥𝐴)
1211ne0d 4300 . 2 ((((𝜑𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦)) → 𝐴 ≠ ∅)
13 tglnne0.1 . . 3 (𝜑𝐴 ∈ ran 𝐿)
141, 2, 3, 4, 13tgisline 26407 . 2 (𝜑 → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝐴 = (𝑥𝐿𝑦) ∧ 𝑥𝑦))
1512, 14r19.29vva 3336 1 (𝜑𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  wne 3016  c0 4290  ran crn 5550  cfv 6349  (class class class)co 7150  Basecbs 16477  TarskiGcstrkg 26210  Itvcitv 26216  LineGclng 26217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-iota 6308  df-fun 6351  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-trkgc 26228  df-trkgb 26229  df-trkgcb 26230  df-trkg 26233
This theorem is referenced by:  hpgerlem  26545
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