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Mirrors > Home > MPE Home > Th. List > tghilberti2 | Structured version Visualization version GIF version |
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.) |
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) |
Ref | Expression |
---|---|
tghilberti2 | ⊢ (𝜑 → ∃*𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) |
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 1135 | . . . . . 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 1135 | . . . . . 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 1135 | . . . . . 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 1135 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ≠ 𝑄) |
12 | simp2l 1201 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 ∈ ran 𝐿) | |
13 | simp3ll 1246 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ∈ 𝑥) | |
14 | simp3lr 1247 | . . . . . 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 26759 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 = (𝑃𝐿𝑄)) |
16 | simp2r 1202 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 ∈ ran 𝐿) | |
17 | simp3rl 1248 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑃 ∈ 𝑦) | |
18 | simp3rr 1249 | . . . . . 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 26759 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑦 = (𝑃𝐿𝑄)) |
20 | 15, 19 | eqtr4d 2782 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿) ∧ ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) → 𝑥 = 𝑦) |
21 | 20 | 3expia 1123 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ran 𝐿 ∧ 𝑦 ∈ ran 𝐿)) → (((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦)) |
22 | 21 | ralrimivva 3115 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐿∀𝑦 ∈ ran 𝐿(((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ∧ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦)) → 𝑥 = 𝑦)) |
23 | eleq2w 2823 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦)) | |
24 | eleq2w 2823 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑄 ∈ 𝑥 ↔ 𝑄 ∈ 𝑦)) | |
25 | 23, 24 | anbi12d 634 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ↔ (𝑃 ∈ 𝑦 ∧ 𝑄 ∈ 𝑦))) |
26 | 25 | rmo4 3660 | . 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 237 | 1 ⊢ (𝜑 → ∃*𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) |
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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