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Theorem tghilberti2 26761
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 (𝜑𝐺 ∈ TarskiG)
tglineelsb2.1 (𝜑𝑃𝐵)
tglineelsb2.2 (𝜑𝑄𝐵)
tglineelsb2.4 (𝜑𝑃𝑄)
Assertion
Ref Expression
tghilberti2 (𝜑 → ∃*𝑥 ∈ ran 𝐿(𝑃𝑥𝑄𝑥))
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.
StepHypRef 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 (𝜑𝐺 ∈ TarskiG)
543ad2ant1 1135 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐿𝑦 ∈ ran 𝐿) ∧ ((𝑃𝑥𝑄𝑥) ∧ (𝑃𝑦𝑄𝑦))) → 𝐺 ∈ TarskiG)
6 tglineelsb2.1 . . . . . . 7 (𝜑𝑃𝐵)
763ad2ant1 1135 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐿𝑦 ∈ ran 𝐿) ∧ ((𝑃𝑥𝑄𝑥) ∧ (𝑃𝑦𝑄𝑦))) → 𝑃𝐵)
8 tglineelsb2.2 . . . . . . 7 (𝜑𝑄𝐵)
983ad2ant1 1135 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐿𝑦 ∈ ran 𝐿) ∧ ((𝑃𝑥𝑄𝑥) ∧ (𝑃𝑦𝑄𝑦))) → 𝑄𝐵)
10 tglineelsb2.4 . . . . . . 7 (𝜑𝑃𝑄)
11103ad2ant1 1135 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐿𝑦 ∈ ran 𝐿) ∧ ((𝑃𝑥𝑄𝑥) ∧ (𝑃𝑦𝑄𝑦))) → 𝑃𝑄)
12 simp2l 1201 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐿𝑦 ∈ ran 𝐿) ∧ ((𝑃𝑥𝑄𝑥) ∧ (𝑃𝑦𝑄𝑦))) → 𝑥 ∈ ran 𝐿)
13 simp3ll 1246 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐿𝑦 ∈ ran 𝐿) ∧ ((𝑃𝑥𝑄𝑥) ∧ (𝑃𝑦𝑄𝑦))) → 𝑃𝑥)
14 simp3lr 1247 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐿𝑦 ∈ ran 𝐿) ∧ ((𝑃𝑥𝑄𝑥) ∧ (𝑃𝑦𝑄𝑦))) → 𝑄𝑥)
151, 2, 3, 5, 7, 9, 11, 11, 12, 13, 14tglinethru 26759 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ran 𝐿𝑦 ∈ ran 𝐿) ∧ ((𝑃𝑥𝑄𝑥) ∧ (𝑃𝑦𝑄𝑦))) → 𝑥 = (𝑃𝐿𝑄))
16 simp2r 1202 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐿𝑦 ∈ ran 𝐿) ∧ ((𝑃𝑥𝑄𝑥) ∧ (𝑃𝑦𝑄𝑦))) → 𝑦 ∈ ran 𝐿)
17 simp3rl 1248 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐿𝑦 ∈ ran 𝐿) ∧ ((𝑃𝑥𝑄𝑥) ∧ (𝑃𝑦𝑄𝑦))) → 𝑃𝑦)
18 simp3rr 1249 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐿𝑦 ∈ ran 𝐿) ∧ ((𝑃𝑥𝑄𝑥) ∧ (𝑃𝑦𝑄𝑦))) → 𝑄𝑦)
191, 2, 3, 5, 7, 9, 11, 11, 16, 17, 18tglinethru 26759 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ran 𝐿𝑦 ∈ ran 𝐿) ∧ ((𝑃𝑥𝑄𝑥) ∧ (𝑃𝑦𝑄𝑦))) → 𝑦 = (𝑃𝐿𝑄))
2015, 19eqtr4d 2782 . . . 4 ((𝜑 ∧ (𝑥 ∈ ran 𝐿𝑦 ∈ ran 𝐿) ∧ ((𝑃𝑥𝑄𝑥) ∧ (𝑃𝑦𝑄𝑦))) → 𝑥 = 𝑦)
21203expia 1123 . . 3 ((𝜑 ∧ (𝑥 ∈ ran 𝐿𝑦 ∈ ran 𝐿)) → (((𝑃𝑥𝑄𝑥) ∧ (𝑃𝑦𝑄𝑦)) → 𝑥 = 𝑦))
2221ralrimivva 3115 . 2 (𝜑 → ∀𝑥 ∈ ran 𝐿𝑦 ∈ ran 𝐿(((𝑃𝑥𝑄𝑥) ∧ (𝑃𝑦𝑄𝑦)) → 𝑥 = 𝑦))
23 eleq2w 2823 . . . 4 (𝑥 = 𝑦 → (𝑃𝑥𝑃𝑦))
24 eleq2w 2823 . . . 4 (𝑥 = 𝑦 → (𝑄𝑥𝑄𝑦))
2523, 24anbi12d 634 . . 3 (𝑥 = 𝑦 → ((𝑃𝑥𝑄𝑥) ↔ (𝑃𝑦𝑄𝑦)))
2625rmo4 3660 . 2 (∃*𝑥 ∈ ran 𝐿(𝑃𝑥𝑄𝑥) ↔ ∀𝑥 ∈ ran 𝐿𝑦 ∈ ran 𝐿(((𝑃𝑥𝑄𝑥) ∧ (𝑃𝑦𝑄𝑦)) → 𝑥 = 𝑦))
2722, 26sylibr 237 1 (𝜑 → ∃*𝑥 ∈ ran 𝐿(𝑃𝑥𝑄𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2112  wne 2943  wral 3064  ∃*wrmo 3067  ran crn 5570  cfv 6401  (class class class)co 7235  Basecbs 16793  TarskiGcstrkg 26553  Itvcitv 26559  LineGclng 26560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-rep 5196  ax-sep 5209  ax-nul 5216  ax-pow 5275  ax-pr 5339  ax-un 7545  ax-cnex 10815  ax-resscn 10816  ax-1cn 10817  ax-icn 10818  ax-addcl 10819  ax-addrcl 10820  ax-mulcl 10821  ax-mulrcl 10822  ax-mulcom 10823  ax-addass 10824  ax-mulass 10825  ax-distr 10826  ax-i2m1 10827  ax-1ne0 10828  ax-1rid 10829  ax-rnegex 10830  ax-rrecex 10831  ax-cnre 10832  ax-pre-lttri 10833  ax-pre-lttrn 10834  ax-pre-ltadd 10835  ax-pre-mulgt0 10836
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4255  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5153  df-tr 5179  df-id 5472  df-eprel 5478  df-po 5486  df-so 5487  df-fr 5527  df-we 5529  df-xp 5575  df-rel 5576  df-cnv 5577  df-co 5578  df-dm 5579  df-rn 5580  df-res 5581  df-ima 5582  df-pred 6179  df-ord 6237  df-on 6238  df-lim 6239  df-suc 6240  df-iota 6359  df-fun 6403  df-fn 6404  df-f 6405  df-f1 6406  df-fo 6407  df-f1o 6408  df-fv 6409  df-riota 7192  df-ov 7238  df-oprab 7239  df-mpo 7240  df-om 7667  df-1st 7783  df-2nd 7784  df-wrecs 8071  df-recs 8132  df-rdg 8170  df-1o 8226  df-oadd 8230  df-er 8415  df-pm 8535  df-en 8651  df-dom 8652  df-sdom 8653  df-fin 8654  df-dju 9547  df-card 9585  df-pnf 10899  df-mnf 10900  df-xr 10901  df-ltxr 10902  df-le 10903  df-sub 11094  df-neg 11095  df-nn 11861  df-2 11923  df-3 11924  df-n0 12121  df-xnn0 12193  df-z 12207  df-uz 12469  df-fz 13126  df-fzo 13269  df-hash 13930  df-word 14103  df-concat 14159  df-s1 14186  df-s2 14446  df-s3 14447  df-trkgc 26571  df-trkgb 26572  df-trkgcb 26573  df-trkg 26576  df-cgrg 26634
This theorem is referenced by:  tglinethrueu  26762
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