Theorem tghilberti2 25754

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 1127	. . . . . 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 1127	. . . . . 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 1127	. . . . . 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 1127	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ≠ 𝑄)
12		simp2l 1241	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 ∈ ran 𝐿)
13		simp3ll 1310	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ∈ 𝑥)
14		simp3lr 1311	. . . . . 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 25752	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 = (𝑃𝐿𝑄))
16		simp2r 1242	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 ∈ ran 𝐿)
17		simp3rl 1312	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ∈ 𝑦)
18		simp3rr 1313	. . . . . 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 25752	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 = (𝑃𝐿𝑄))
20	15, 19	eqtr4d 2808	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 = 𝑦)
21	20	3expia 1114	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿)) → (((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦))
22	21	ralrimivva 3120	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐿∀𝑦 ∈ ran 𝐿(((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦))
23		eleq2w 2834	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
24		eleq2w 2834	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑄 ∈ 𝑥 ↔ 𝑄 ∈ 𝑦))
25	23, 24	anbi12d 616	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ↔ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)))
26	25	rmo4 3551	. 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 224	1 ⊢ (𝜑 → ∃*𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))

Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ≠ wne 2943  ∀wral 3061  ∃*wrmo 3064  ran crn 5250  ‘cfv 6031  (class class class)co 6793  Basecbs 16064  TarskiGcstrkg 25550  Itvcitv 25556  LineGclng 25557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-xnn0 11566  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-concat 13497  df-s1 13498  df-s2 13802  df-s3 13803  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573  df-cgrg 25627

This theorem is referenced by:  tglinethrueu  25755