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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
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