Theorem tghilberti2 28646

Description: There is at most one line through any two distinct points. Hilbert's axiom I.2 for geometry. (Contributed by Thierry Arnoux, 25-May-2019.)

Hypotheses

Ref	Expression
tglineelsb2.p	⊢ 𝐵 = (Base‘𝐺)
tglineelsb2.i	⊢ 𝐼 = (Itv‘𝐺)
tglineelsb2.l	⊢ 𝐿 = (LineG‘𝐺)
tglineelsb2.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglineelsb2.1	⊢ (𝜑 → 𝑃 ∈ 𝐵)
tglineelsb2.2	⊢ (𝜑 → 𝑄 ∈ 𝐵)
tglineelsb2.4	⊢ (𝜑 → 𝑃 ≠ 𝑄)

Assertion

Ref	Expression
tghilberti2	⊢ (𝜑 → ∃*𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐺   𝑥,𝐼   𝑥,𝐿   𝑥,𝑃   𝑥,𝑄   𝜑,𝑥

Proof of Theorem tghilberti2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tglineelsb2.p	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
2		tglineelsb2.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
3		tglineelsb2.l	. . . . . 6 ⊢ 𝐿 = (LineG‘𝐺)
4		tglineelsb2.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝐺 ∈ TarskiG)
6		tglineelsb2.1	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
7	6	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ∈ 𝐵)
8		tglineelsb2.2	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
9	8	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑄 ∈ 𝐵)
10		tglineelsb2.4	. . . . . . 7 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
11	10	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ≠ 𝑄)
12		simp2l 1200	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 ∈ ran 𝐿)
13		simp3ll 1245	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ∈ 𝑥)
14		simp3lr 1246	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑄 ∈ 𝑥)
15	1, 2, 3, 5, 7, 9, 11, 11, 12, 13, 14	tglinethru 28644	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 = (𝑃𝐿𝑄))
16		simp2r 1201	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 ∈ ran 𝐿)
17		simp3rl 1247	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ∈ 𝑦)
18		simp3rr 1248	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑄 ∈ 𝑦)
19	1, 2, 3, 5, 7, 9, 11, 11, 16, 17, 18	tglinethru 28644	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 = (𝑃𝐿𝑄))
20	15, 19	eqtr4d 2780	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 = 𝑦)
21	20	3expia 1122	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿)) → (((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦))
22	21	ralrimivva 3202	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐿∀𝑦 ∈ ran 𝐿(((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦))
23		eleq2w 2825	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
24		eleq2w 2825	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑄 ∈ 𝑥 ↔ 𝑄 ∈ 𝑦))
25	23, 24	anbi12d 632	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ↔ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)))
26	25	rmo4 3736	. 2 ⊢ (∃*𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ↔ ∀𝑥 ∈ ran 𝐿∀𝑦 ∈ ran 𝐿(((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦))
27	22, 26	sylibr 234	1 ⊢ (𝜑 → ∃*𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃*wrmo 3379  ran crn 5686  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  TarskiGcstrkg 28435  Itvcitv 28441  LineGclng 28442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-s2 14887  df-s3 14888  df-trkgc 28456  df-trkgb 28457  df-trkgcb 28458  df-trkg 28461  df-cgrg 28519

This theorem is referenced by:  tglinethrueu  28647