Theorem tghilberti2 28664

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 1133	. . . . . 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 1133	. . . . . 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 1133	. . . . . 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 1133	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ≠ 𝑄)
12		simp2l 1199	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 ∈ ran 𝐿)
13		simp3ll 1244	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ∈ 𝑥)
14		simp3lr 1245	. . . . . 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 28662	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 = (𝑃𝐿𝑄))
16		simp2r 1200	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 ∈ ran 𝐿)
17		simp3rl 1246	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ∈ 𝑦)
18		simp3rr 1247	. . . . . 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 28662	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 = (𝑃𝐿𝑄))
20	15, 19	eqtr4d 2783	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 = 𝑦)
21	20	3expia 1121	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿)) → (((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦))
22	21	ralrimivva 3208	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐿∀𝑦 ∈ ran 𝐿(((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦))
23		eleq2w 2828	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
24		eleq2w 2828	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑄 ∈ 𝑥 ↔ 𝑄 ∈ 𝑦))
25	23, 24	anbi12d 631	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ↔ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)))
26	25	rmo4 3752	. 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 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃*wrmo 3387  ran crn 5701  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  TarskiGcstrkg 28453  Itvcitv 28459  LineGclng 28460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-er 8763  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-concat 14619  df-s1 14644  df-s2 14897  df-s3 14898  df-trkgc 28474  df-trkgb 28475  df-trkgcb 28476  df-trkg 28479  df-cgrg 28537

This theorem is referenced by:  tglinethrueu  28665