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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hilbert1.1 | Structured version Visualization version GIF version |
Description: There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
hilbert1.1 | ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → ∃𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1172 | . . . . 5 ⊢ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ (𝔼‘𝑁)) | |
2 | simp2 1173 | . . . . 5 ⊢ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ (𝔼‘𝑁)) | |
3 | simp3 1174 | . . . . 5 ⊢ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) → 𝑃 ≠ 𝑄) | |
4 | eqidd 2826 | . . . . 5 ⊢ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) → (𝑃Line𝑄) = (𝑃Line𝑄)) | |
5 | neeq1 3061 | . . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝 ≠ 𝑞 ↔ 𝑃 ≠ 𝑞)) | |
6 | oveq1 6912 | . . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑝Line𝑞) = (𝑃Line𝑞)) | |
7 | 6 | eqeq2d 2835 | . . . . . . 7 ⊢ (𝑝 = 𝑃 → ((𝑃Line𝑄) = (𝑝Line𝑞) ↔ (𝑃Line𝑄) = (𝑃Line𝑞))) |
8 | 5, 7 | anbi12d 626 | . . . . . 6 ⊢ (𝑝 = 𝑃 → ((𝑝 ≠ 𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞)) ↔ (𝑃 ≠ 𝑞 ∧ (𝑃Line𝑄) = (𝑃Line𝑞)))) |
9 | neeq2 3062 | . . . . . . 7 ⊢ (𝑞 = 𝑄 → (𝑃 ≠ 𝑞 ↔ 𝑃 ≠ 𝑄)) | |
10 | oveq2 6913 | . . . . . . . 8 ⊢ (𝑞 = 𝑄 → (𝑃Line𝑞) = (𝑃Line𝑄)) | |
11 | 10 | eqeq2d 2835 | . . . . . . 7 ⊢ (𝑞 = 𝑄 → ((𝑃Line𝑄) = (𝑃Line𝑞) ↔ (𝑃Line𝑄) = (𝑃Line𝑄))) |
12 | 9, 11 | anbi12d 626 | . . . . . 6 ⊢ (𝑞 = 𝑄 → ((𝑃 ≠ 𝑞 ∧ (𝑃Line𝑄) = (𝑃Line𝑞)) ↔ (𝑃 ≠ 𝑄 ∧ (𝑃Line𝑄) = (𝑃Line𝑄)))) |
13 | 8, 12 | rspc2ev 3541 | . . . . 5 ⊢ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃Line𝑄) = (𝑃Line𝑄))) → ∃𝑝 ∈ (𝔼‘𝑁)∃𝑞 ∈ (𝔼‘𝑁)(𝑝 ≠ 𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞))) |
14 | 1, 2, 3, 4, 13 | syl112anc 1499 | . . . 4 ⊢ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) → ∃𝑝 ∈ (𝔼‘𝑁)∃𝑞 ∈ (𝔼‘𝑁)(𝑝 ≠ 𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞))) |
15 | fveq2 6433 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁)) | |
16 | 15 | rexeqdv 3357 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞)) ↔ ∃𝑞 ∈ (𝔼‘𝑁)(𝑝 ≠ 𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞)))) |
17 | 15, 16 | rexeqbidv 3365 | . . . . 5 ⊢ (𝑛 = 𝑁 → (∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞)) ↔ ∃𝑝 ∈ (𝔼‘𝑁)∃𝑞 ∈ (𝔼‘𝑁)(𝑝 ≠ 𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞)))) |
18 | 17 | rspcev 3526 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ ∃𝑝 ∈ (𝔼‘𝑁)∃𝑞 ∈ (𝔼‘𝑁)(𝑝 ≠ 𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞))) → ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞))) |
19 | 14, 18 | sylan2 588 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞))) |
20 | ellines 32798 | . . 3 ⊢ ((𝑃Line𝑄) ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ (𝑃Line𝑄) = (𝑝Line𝑞))) | |
21 | 19, 20 | sylibr 226 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → (𝑃Line𝑄) ∈ LinesEE) |
22 | linerflx1 32795 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → 𝑃 ∈ (𝑃Line𝑄)) | |
23 | linerflx2 32797 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → 𝑄 ∈ (𝑃Line𝑄)) | |
24 | eleq2 2895 | . . . 4 ⊢ (𝑥 = (𝑃Line𝑄) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑃Line𝑄))) | |
25 | eleq2 2895 | . . . 4 ⊢ (𝑥 = (𝑃Line𝑄) → (𝑄 ∈ 𝑥 ↔ 𝑄 ∈ (𝑃Line𝑄))) | |
26 | 24, 25 | anbi12d 626 | . . 3 ⊢ (𝑥 = (𝑃Line𝑄) → ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ↔ (𝑃 ∈ (𝑃Line𝑄) ∧ 𝑄 ∈ (𝑃Line𝑄)))) |
27 | 26 | rspcev 3526 | . 2 ⊢ (((𝑃Line𝑄) ∈ LinesEE ∧ (𝑃 ∈ (𝑃Line𝑄) ∧ 𝑄 ∈ (𝑃Line𝑄))) → ∃𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) |
28 | 21, 22, 23, 27 | syl12anc 872 | 1 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → ∃𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ∃wrex 3118 ‘cfv 6123 (class class class)co 6905 ℕcn 11350 𝔼cee 26187 Linecline2 32780 LinesEEclines2 32782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-ec 8011 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-rp 12113 df-ico 12469 df-icc 12470 df-fz 12620 df-fzo 12761 df-seq 13096 df-exp 13155 df-hash 13411 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-clim 14596 df-sum 14794 df-ee 26190 df-btwn 26191 df-cgr 26192 df-colinear 32685 df-line2 32783 df-lines2 32785 |
This theorem is referenced by: linethrueu 32802 |
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