Theorem tglnne 28696

Description: It takes two different points to form a line. (Contributed by Thierry Arnoux, 27-Nov-2019.)

Hypotheses

Ref	Expression
tglineelsb2.p	⊢ 𝐵 = (Base‘𝐺)
tglineelsb2.i	⊢ 𝐼 = (Itv‘𝐺)
tglineelsb2.l	⊢ 𝐿 = (LineG‘𝐺)
tglineelsb2.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglnne.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
tglnne.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
tglnne.1	⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)

Assertion

Ref	Expression
tglnne	⊢ (𝜑 → 𝑋 ≠ 𝑌)

Proof of Theorem tglnne

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

Step	Hyp	Ref	Expression
1		tglineelsb2.p	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		tglineelsb2.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
3		tglineelsb2.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		tglineelsb2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 731	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐺 ∈ TarskiG)
6		tglnne.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7	6	ad3antrrr 731	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑋 ∈ 𝐵)
8		tglnne.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	8	ad3antrrr 731	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑌 ∈ 𝐵)
10		simpllr 776	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐵)
11		simplr 769	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐵)
12		simprr 773	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
13		eqid 2736	. . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺)
14	1, 13, 3, 5, 10, 11	tgbtwntriv1 28559	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑥𝐼𝑦))
15	1, 3, 2, 5, 10, 11, 10, 12, 14	btwnlng1 28687	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦))
16		simprl 771	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → (𝑋𝐿𝑌) = (𝑥𝐿𝑦))
17	15, 16	eleqtrrd 2839	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑋𝐿𝑌))
18	1, 2, 3, 5, 7, 9, 17	tglngne 28618	. 2 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑋 ≠ 𝑌)
19		tglnne.1	. . 3 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿)
20	1, 3, 2, 4, 19	tgisline 28695	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦))
21	18, 20	r19.29vva 3197	1 ⊢ (𝜑 → 𝑋 ≠ 𝑌)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ran crn 5632  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  distcds 17229  TarskiGcstrkg 28495  Itvcitv 28501  LineGclng 28502

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 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-trkgc 28516  df-trkgb 28517  df-trkgcb 28518  df-trkg 28521

This theorem is referenced by:  footne  28791  footeq  28792  hlperpnel  28793  colperp  28797  mideulem2  28802  opphllem  28803  midex  28805  opphllem3  28817  opphllem6  28820  opphl  28822  lmieu  28852  lnperpex  28871  trgcopy  28872