Theorem tglnne 29096

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 743	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐺 ∈ TarskiG)
6		tglnne.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7	6	ad3antrrr 743	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑋 ∈ 𝐵)
8		tglnne.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	8	ad3antrrr 743	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑌 ∈ 𝐵)
10		simpllr 788	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐵)
11		simplr 781	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐵)
12		simprr 785	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
13		eqid 2761	. . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺)
14	1, 13, 3, 5, 10, 11	tgbtwntriv1 28954	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑥𝐼𝑦))
15	1, 3, 2, 5, 10, 11, 10, 12, 14	btwnlng1 29087	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦))
16		simprl 783	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → (𝑋𝐿𝑌) = (𝑥𝐿𝑦))
17	15, 16	eleqtrrd 2864	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑋𝐿𝑌))
18	1, 2, 3, 5, 7, 9, 17	tglngne 29013	. 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 29095	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦))
21	18, 20	r19.29vva 3223	1 ⊢ (𝜑 → 𝑋 ≠ 𝑌)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ran crn 5652  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  distcds 17437  TarskiGcstrkg 28889  Itvcitv 28895  LineGclng 28896

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-trkgc 28910  df-trkgb 28911  df-trkgcb 28912  df-trkg 28915

This theorem is used by:  footne  29198  footeq  29199  hlperpnel  29201  colperp  29205  mideulem2  29210  opphllem  29211  midex  29213  opphllem3  29225  opphllem6  29228  opphl  29230  lmieu  29289  lnperpex  29309  trgcopy  29311  perpeqlem  29347  perpeq  29348  prlngex  29429  prlngsymquadlem  29441  prlngsymquadopp  29443  quadcgrprlng  29444  tgaltai  29445