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Mirrors > Home > MPE Home > Th. List > tghilberti1 | 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 Thierry Arnoux, 25-May-2019.) |
Ref | Expression |
---|---|
tglineelsb2.p | β’ π΅ = (BaseβπΊ) |
tglineelsb2.i | β’ πΌ = (ItvβπΊ) |
tglineelsb2.l | β’ πΏ = (LineGβπΊ) |
tglineelsb2.g | β’ (π β πΊ β TarskiG) |
tglineelsb2.1 | β’ (π β π β π΅) |
tglineelsb2.2 | β’ (π β π β π΅) |
tglineelsb2.4 | β’ (π β π β π) |
Ref | Expression |
---|---|
tghilberti1 | β’ (π β βπ₯ β ran πΏ(π β π₯ β§ π β π₯)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglineelsb2.p | . . 3 β’ π΅ = (BaseβπΊ) | |
2 | tglineelsb2.i | . . 3 β’ πΌ = (ItvβπΊ) | |
3 | tglineelsb2.l | . . 3 β’ πΏ = (LineGβπΊ) | |
4 | tglineelsb2.g | . . 3 β’ (π β πΊ β TarskiG) | |
5 | tglineelsb2.1 | . . 3 β’ (π β π β π΅) | |
6 | tglineelsb2.2 | . . 3 β’ (π β π β π΅) | |
7 | tglineelsb2.4 | . . 3 β’ (π β π β π) | |
8 | 1, 2, 3, 4, 5, 6, 7 | tgelrnln 27878 | . 2 β’ (π β (ππΏπ) β ran πΏ) |
9 | 1, 2, 3, 4, 5, 6, 7 | tglinerflx1 27881 | . 2 β’ (π β π β (ππΏπ)) |
10 | 1, 2, 3, 4, 5, 6, 7 | tglinerflx2 27882 | . 2 β’ (π β π β (ππΏπ)) |
11 | eleq2 2822 | . . . 4 β’ (π₯ = (ππΏπ) β (π β π₯ β π β (ππΏπ))) | |
12 | eleq2 2822 | . . . 4 β’ (π₯ = (ππΏπ) β (π β π₯ β π β (ππΏπ))) | |
13 | 11, 12 | anbi12d 631 | . . 3 β’ (π₯ = (ππΏπ) β ((π β π₯ β§ π β π₯) β (π β (ππΏπ) β§ π β (ππΏπ)))) |
14 | 13 | rspcev 3612 | . 2 β’ (((ππΏπ) β ran πΏ β§ (π β (ππΏπ) β§ π β (ππΏπ))) β βπ₯ β ran πΏ(π β π₯ β§ π β π₯)) |
15 | 8, 9, 10, 14 | syl12anc 835 | 1 β’ (π β βπ₯ β ran πΏ(π β π₯ β§ π β π₯)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β wne 2940 βwrex 3070 ran crn 5677 βcfv 6543 (class class class)co 7408 Basecbs 17143 TarskiGcstrkg 27675 Itvcitv 27681 LineGclng 27682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-trkgc 27696 df-trkgb 27697 df-trkgcb 27698 df-trkg 27701 |
This theorem is referenced by: tglinethrueu 27887 |
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