Theorem tglngne 26338

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglngval.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
tglngval.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
tglngne.1	⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Assertion

Ref	Expression
tglngne	⊢ (𝜑 → 𝑋 ≠ 𝑌)

Proof of Theorem tglngne

Step	Hyp	Ref	Expression
1		tglngne.1	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))
2		df-ov 7161	. . . . . 6 ⊢ (𝑋𝐿𝑌) = (𝐿‘⟨𝑋, 𝑌⟩)
3	1, 2	eleqtrdi 2925	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝐿‘⟨𝑋, 𝑌⟩))
4		elfvdm 6704	. . . . 5 ⊢ (𝑍 ∈ (𝐿‘⟨𝑋, 𝑌⟩) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐿)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom 𝐿)
6		tglngval.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		tglngval.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
8		tglngval.l	. . . . . 6 ⊢ 𝐿 = (LineG‘𝐺)
9		tglngval.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
10	7, 8, 9	tglnfn 26335	. . . . 5 ⊢ (𝐺 ∈ TarskiG → 𝐿 Fn ((𝑃 × 𝑃) ∖ I ))
11		fndm 6457	. . . . 5 ⊢ (𝐿 Fn ((𝑃 × 𝑃) ∖ I ) → dom 𝐿 = ((𝑃 × 𝑃) ∖ I ))
12	6, 10, 11	3syl 18	. . . 4 ⊢ (𝜑 → dom 𝐿 = ((𝑃 × 𝑃) ∖ I ))
13	5, 12	eleqtrd 2917	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ ((𝑃 × 𝑃) ∖ I ))
14	13	eldifbd 3951	. 2 ⊢ (𝜑 → ¬ ⟨𝑋, 𝑌⟩ ∈ I )
15		df-br 5069	. . . 4 ⊢ (𝑋 I 𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ I )
16		tglngval.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
17		ideqg 5724	. . . . 5 ⊢ (𝑌 ∈ 𝑃 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌))
18	16, 17	syl 17	. . . 4 ⊢ (𝜑 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌))
19	15, 18	syl5bbr 287	. . 3 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 = 𝑌))
20	19	necon3bbid 3055	. 2 ⊢ (𝜑 → (¬ ⟨𝑋, 𝑌⟩ ∈ I ↔ 𝑋 ≠ 𝑌))
21	14, 20	mpbid 234	1 ⊢ (𝜑 → 𝑋 ≠ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1537   ∈ wcel 2114   ≠ wne 3018   ∖ cdif 3935  ⟨cop 4575   class class class wbr 5068   I cid 5461   × cxp 5555  dom cdm 5557   Fn wfn 6352  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  TarskiGcstrkg 26218  Itvcitv 26224  LineGclng 26225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692  df-trkg 26241

This theorem is referenced by:  lnhl  26403  tglnne  26416  tglineneq  26432  tglineinteq  26433  ncolncol  26434  coltr  26435  coltr3  26436  perprag  26514