Theorem tglnne 25996

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 720	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐺 ∈ TarskiG)
6		tglnne.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7	6	ad3antrrr 720	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑋 ∈ 𝐵)
8		tglnne.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	8	ad3antrrr 720	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑌 ∈ 𝐵)
10		simpllr 766	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐵)
11		simplr 759	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐵)
12		simprr 763	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
13		eqid 2778	. . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺)
14	1, 13, 3, 5, 10, 11	tgbtwntriv1 25859	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑥𝐼𝑦))
15	1, 3, 2, 5, 10, 11, 10, 12, 14	btwnlng1 25987	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦))
16		simprl 761	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → (𝑋𝐿𝑌) = (𝑥𝐿𝑦))
17	15, 16	eleqtrrd 2862	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑋𝐿𝑌))
18	1, 2, 3, 5, 7, 9, 17	tglngne 25918	. 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 25995	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦))
21	18, 20	r19.29vva 3267	1 ⊢ (𝜑 → 𝑋 ≠ 𝑌)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ran crn 5358  ‘cfv 6137  (class class class)co 6924  Basecbs 16266  distcds 16358  TarskiGcstrkg 25798  Itvcitv 25804  LineGclng 25805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-trkgc 25816  df-trkgb 25817  df-trkgcb 25818  df-trkg 25821

This theorem is referenced by:  footne  26088  footeq  26089  hlperpnel  26090  colperp  26094  mideulem2  26099  opphllem  26100  midex  26102  opphllem3  26114  opphllem6  26117  opphl  26119  lmieu  26149  lnperpex  26168  trgcopy  26169