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Mirrors > Home > MPE Home > Th. List > tglnssp | Structured version Visualization version GIF version |
Description: Lines are subset of the geometry base set. That is, lines are sets of points. (Contributed by Thierry Arnoux, 17-May-2019.) |
Ref | Expression |
---|---|
tglngval.p | β’ π = (BaseβπΊ) |
tglngval.l | β’ πΏ = (LineGβπΊ) |
tglngval.i | β’ πΌ = (ItvβπΊ) |
tglngval.g | β’ (π β πΊ β TarskiG) |
tglngval.x | β’ (π β π β π) |
tglngval.y | β’ (π β π β π) |
tglngval.z | β’ (π β π β π) |
Ref | Expression |
---|---|
tglnssp | β’ (π β (ππΏπ) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglngval.p | . . 3 β’ π = (BaseβπΊ) | |
2 | tglngval.l | . . 3 β’ πΏ = (LineGβπΊ) | |
3 | tglngval.i | . . 3 β’ πΌ = (ItvβπΊ) | |
4 | tglngval.g | . . 3 β’ (π β πΊ β TarskiG) | |
5 | tglngval.x | . . 3 β’ (π β π β π) | |
6 | tglngval.y | . . 3 β’ (π β π β π) | |
7 | tglngval.z | . . 3 β’ (π β π β π) | |
8 | 1, 2, 3, 4, 5, 6, 7 | tglngval 28373 | . 2 β’ (π β (ππΏπ) = {π§ β π β£ (π§ β (ππΌπ) β¨ π β (π§πΌπ) β¨ π β (ππΌπ§))}) |
9 | ssrab2 4075 | . 2 β’ {π§ β π β£ (π§ β (ππΌπ) β¨ π β (π§πΌπ) β¨ π β (ππΌπ§))} β π | |
10 | 8, 9 | eqsstrdi 4034 | 1 β’ (π β (ππΏπ) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β¨ w3o 1083 = wceq 1533 β wcel 2098 β wne 2936 {crab 3428 β wss 3947 βcfv 6551 (class class class)co 7424 Basecbs 17185 TarskiGcstrkg 28249 Itvcitv 28255 LineGclng 28256 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-iota 6503 df-fun 6553 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-trkg 28275 |
This theorem is referenced by: tglineelsb2 28454 tglinecom 28457 |
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