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Theorem tglngne 26388
 Description: It takes two different points to form a line. (Contributed by Thierry Arnoux, 6-Aug-2019.)
Hypotheses
Ref Expression
tglngval.p 𝑃 = (Base‘𝐺)
tglngval.l 𝐿 = (LineG‘𝐺)
tglngval.i 𝐼 = (Itv‘𝐺)
tglngval.g (𝜑𝐺 ∈ TarskiG)
tglngval.x (𝜑𝑋𝑃)
tglngval.y (𝜑𝑌𝑃)
tglngne.1 (𝜑𝑍 ∈ (𝑋𝐿𝑌))
Assertion
Ref Expression
tglngne (𝜑𝑋𝑌)

Proof of Theorem tglngne
StepHypRef Expression
1 tglngne.1 . . . . . 6 (𝜑𝑍 ∈ (𝑋𝐿𝑌))
2 df-ov 7148 . . . . . 6 (𝑋𝐿𝑌) = (𝐿‘⟨𝑋, 𝑌⟩)
31, 2eleqtrdi 2900 . . . . 5 (𝜑𝑍 ∈ (𝐿‘⟨𝑋, 𝑌⟩))
4 elfvdm 6687 . . . . 5 (𝑍 ∈ (𝐿‘⟨𝑋, 𝑌⟩) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐿)
53, 4syl 17 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom 𝐿)
6 tglngval.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
7 tglngval.p . . . . . 6 𝑃 = (Base‘𝐺)
8 tglngval.l . . . . . 6 𝐿 = (LineG‘𝐺)
9 tglngval.i . . . . . 6 𝐼 = (Itv‘𝐺)
107, 8, 9tglnfn 26385 . . . . 5 (𝐺 ∈ TarskiG → 𝐿 Fn ((𝑃 × 𝑃) ∖ I ))
11 fndm 6433 . . . . 5 (𝐿 Fn ((𝑃 × 𝑃) ∖ I ) → dom 𝐿 = ((𝑃 × 𝑃) ∖ I ))
126, 10, 113syl 18 . . . 4 (𝜑 → dom 𝐿 = ((𝑃 × 𝑃) ∖ I ))
135, 12eleqtrd 2892 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((𝑃 × 𝑃) ∖ I ))
1413eldifbd 3896 . 2 (𝜑 → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
15 df-br 5035 . . . 4 (𝑋 I 𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ I )
16 tglngval.y . . . . 5 (𝜑𝑌𝑃)
17 ideqg 5690 . . . . 5 (𝑌𝑃 → (𝑋 I 𝑌𝑋 = 𝑌))
1816, 17syl 17 . . . 4 (𝜑 → (𝑋 I 𝑌𝑋 = 𝑌))
1915, 18bitr3id 288 . . 3 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 = 𝑌))
2019necon3bbid 3024 . 2 (𝜑 → (¬ ⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋𝑌))
2114, 20mpbid 235 1 (𝜑𝑋𝑌)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   ∖ cdif 3880  ⟨cop 4534   class class class wbr 5034   I cid 5428   × cxp 5521  dom cdm 5523   Fn wfn 6327  ‘cfv 6332  (class class class)co 7145  Basecbs 16495  TarskiGcstrkg 26268  Itvcitv 26274  LineGclng 26275 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7684  df-2nd 7685  df-trkg 26291 This theorem is referenced by:  lnhl  26453  tglnne  26466  tglineneq  26482  tglineinteq  26483  ncolncol  26484  coltr  26485  coltr3  26486  perprag  26564
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