Theorem tglnne0 28543

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 2729	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2729	. . . . 5 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
3		tglnne0.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
4		tglnne0.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 730	. . . . 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 28536	. . . 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 2831	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐴)
12	11	ne0d 4301	. 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 28530	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦))
15	12, 14	r19.29vva 3195	1 ⊢ (𝜑 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∅c0 4292  ran crn 5632  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  TarskiGcstrkg 28330  Itvcitv 28336  LineGclng 28337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-trkgc 28351  df-trkgb 28352  df-trkgcb 28353  df-trkg 28356

This theorem is referenced by:  hpgerlem  28668