Theorem tghilberti2 26427

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 1129	. . . . . 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 1129	. . . . . 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 1129	. . . . . 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 1129	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ≠ 𝑄)
12		simp2l 1195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 ∈ ran 𝐿)
13		simp3ll 1240	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ∈ 𝑥)
14		simp3lr 1241	. . . . . 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 26425	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 = (𝑃𝐿𝑄))
16		simp2r 1196	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 ∈ ran 𝐿)
17		simp3rl 1242	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ∈ 𝑦)
18		simp3rr 1243	. . . . . 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 26425	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 = (𝑃𝐿𝑄))
20	15, 19	eqtr4d 2862	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 = 𝑦)
21	20	3expia 1117	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿)) → (((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦))
22	21	ralrimivva 3194	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐿∀𝑦 ∈ ran 𝐿(((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦))
23		eleq2w 2899	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
24		eleq2w 2899	. . . 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 3724	. 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 236	1 ⊢ (𝜑 → ∃*𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113   ≠ wne 3019  ∀wral 3141  ∃*wrmo 3144  ran crn 5559  ‘cfv 6358  (class class class)co 7159  Basecbs 16486  TarskiGcstrkg 26219  Itvcitv 26225  LineGclng 26226

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-pm 8412  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-dju 9333  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-2 11703  df-3 11704  df-n0 11901  df-xnn0 11971  df-z 11985  df-uz 12247  df-fz 12896  df-fzo 13037  df-hash 13694  df-word 13865  df-concat 13926  df-s1 13953  df-s2 14213  df-s3 14214  df-trkgc 26237  df-trkgb 26238  df-trkgcb 26239  df-trkg 26242  df-cgrg 26300

This theorem is referenced by:  tglinethrueu  26428