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Mirrors > Home > MPE Home > Th. List > pthontrlon | Structured version Visualization version GIF version |
Description: A path between two vertices is a trail between these vertices. (Contributed by AV, 24-Jan-2021.) |
Ref | Expression |
---|---|
pthontrlon | โข (๐น(๐ด(PathsOnโ๐บ)๐ต)๐ โ ๐น(๐ด(TrailsOnโ๐บ)๐ต)๐) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 โข (Vtxโ๐บ) = (Vtxโ๐บ) | |
2 | 1 | pthsonprop 28157 | . 2 โข (๐น(๐ด(PathsOnโ๐บ)๐ต)๐ โ ((๐บ โ V โง ๐ด โ (Vtxโ๐บ) โง ๐ต โ (Vtxโ๐บ)) โง (๐น โ V โง ๐ โ V) โง (๐น(๐ด(TrailsOnโ๐บ)๐ต)๐ โง ๐น(Pathsโ๐บ)๐))) |
3 | simp3l 1201 | . 2 โข (((๐บ โ V โง ๐ด โ (Vtxโ๐บ) โง ๐ต โ (Vtxโ๐บ)) โง (๐น โ V โง ๐ โ V) โง (๐น(๐ด(TrailsOnโ๐บ)๐ต)๐ โง ๐น(Pathsโ๐บ)๐)) โ ๐น(๐ด(TrailsOnโ๐บ)๐ต)๐) | |
4 | 2, 3 | syl 17 | 1 โข (๐น(๐ด(PathsOnโ๐บ)๐ต)๐ โ ๐น(๐ด(TrailsOnโ๐บ)๐ต)๐) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 397 โง w3a 1087 โ wcel 2104 Vcvv 3437 class class class wbr 5081 โcfv 6458 (class class class)co 7307 Vtxcvtx 27411 TrailsOnctrlson 28104 Pathscpths 28125 PathsOncpthson 28127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5500 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-ov 7310 df-oprab 7311 df-mpo 7312 df-1st 7863 df-2nd 7864 df-pthson 28131 |
This theorem is referenced by: uhgrwkspth 28168 usgr2wlkspth 28172 wspthneq1eq2 28270 conngrv2edg 28604 |
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