Theorem tglnne 28452

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 728	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐺 ∈ TarskiG)
6		tglnne.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7	6	ad3antrrr 728	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑋 ∈ 𝐵)
8		tglnne.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	8	ad3antrrr 728	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑌 ∈ 𝐵)
10		simpllr 774	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐵)
11		simplr 767	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐵)
12		simprr 771	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
13		eqid 2728	. . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺)
14	1, 13, 3, 5, 10, 11	tgbtwntriv1 28315	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑥𝐼𝑦))
15	1, 3, 2, 5, 10, 11, 10, 12, 14	btwnlng1 28443	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦))
16		simprl 769	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → (𝑋𝐿𝑌) = (𝑥𝐿𝑦))
17	15, 16	eleqtrrd 2832	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑋𝐿𝑌))
18	1, 2, 3, 5, 7, 9, 17	tglngne 28374	. 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 28451	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦))
21	18, 20	r19.29vva 3211	1 ⊢ (𝜑 → 𝑋 ≠ 𝑌)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ≠ wne 2937  ran crn 5683  ‘cfv 6553  (class class class)co 7426  Basecbs 17187  distcds 17249  TarskiGcstrkg 28251  Itvcitv 28257  LineGclng 28258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-trkgc 28272  df-trkgb 28273  df-trkgcb 28274  df-trkg 28277

This theorem is referenced by:  footne  28547  footeq  28548  hlperpnel  28549  colperp  28553  mideulem2  28558  opphllem  28559  midex  28561  opphllem3  28573  opphllem6  28576  opphl  28578  lmieu  28608  lnperpex  28627  trgcopy  28628