Theorem tglnne0 28726

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 2737	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2737	. . . . 5 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
3		tglnne0.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
4		tglnne0.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 731	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐺 ∈ TarskiG)
6		simpllr 776	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (Base‘𝐺))
7		simplr 769	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ (Base‘𝐺))
8		simprr 773	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
9	1, 2, 3, 5, 6, 7, 8	tglinerflx1 28719	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦))
10		simprl 771	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐴 = (𝑥𝐿𝑦))
11	9, 10	eleqtrrd 2840	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐴)
12	11	ne0d 4283	. 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 28713	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦))
15	12, 14	r19.29vva 3198	1 ⊢ (𝜑 → 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∅c0 4274  ran crn 5627  ‘cfv 6494  (class class class)co 7362  Basecbs 17174  TarskiGcstrkg 28513  Itvcitv 28519  LineGclng 28520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-iota 6450  df-fun 6496  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-trkgc 28534  df-trkgb 28535  df-trkgcb 28536  df-trkg 28539

This theorem is referenced by:  hpgerlem  28851