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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 27542 | . 2 β’ (π β (ππΏπ) = {π§ β π β£ (π§ β (ππΌπ) β¨ π β (π§πΌπ) β¨ π β (ππΌπ§))}) |
9 | ssrab2 4041 | . 2 β’ {π§ β π β£ (π§ β (ππΌπ) β¨ π β (π§πΌπ) β¨ π β (ππΌπ§))} β π | |
10 | 8, 9 | eqsstrdi 4002 | 1 β’ (π β (ππΏπ) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β¨ w3o 1087 = wceq 1542 β wcel 2107 β wne 2940 {crab 3406 β wss 3914 βcfv 6500 (class class class)co 7361 Basecbs 17091 TarskiGcstrkg 27418 Itvcitv 27424 LineGclng 27425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-trkg 27444 |
This theorem is referenced by: tglineelsb2 27623 tglinecom 27626 |
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