Theorem tglnne0 28666

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglnne0.1	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)

Assertion

Ref	Expression
tglnne0	⊢ (𝜑 → 𝐴 ≠ ∅)

Proof of Theorem tglnne0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2740	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2740	. . . . 5 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
3		tglnne0.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
4		tglnne0.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 729	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐺 ∈ TarskiG)
6		simpllr 775	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (Base‘𝐺))
7		simplr 768	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ (Base‘𝐺))
8		simprr 772	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
9	1, 2, 3, 5, 6, 7, 8	tglinerflx1 28659	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦))
10		simprl 770	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐴 = (𝑥𝐿𝑦))
11	9, 10	eleqtrrd 2847	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐴)
12	11	ne0d 4365	. 2 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐴 ≠ ∅)
13		tglnne0.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
14	1, 2, 3, 4, 13	tgisline 28653	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦))
15	12, 14	r19.29vva 3222	1 ⊢ (𝜑 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∅c0 4352  ran crn 5701  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  TarskiGcstrkg 28453  Itvcitv 28459  LineGclng 28460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-trkgc 28474  df-trkgb 28475  df-trkgcb 28476  df-trkg 28479

This theorem is referenced by:  hpgerlem  28791