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Mirrors > Home > MPE Home > Th. List > clwwlknon1sn | Structured version Visualization version GIF version |
Description: The set of (closed) walks on vertex π of length 1 as words over the set of vertices is a singleton containing the singleton word consisting of π iff there is a loop at π. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.) |
Ref | Expression |
---|---|
clwwlknon1.v | β’ π = (VtxβπΊ) |
clwwlknon1.c | β’ πΆ = (ClWWalksNOnβπΊ) |
clwwlknon1.e | β’ πΈ = (EdgβπΊ) |
Ref | Expression |
---|---|
clwwlknon1sn | β’ (π β π β ((ππΆ1) = {β¨βπββ©} β {π} β πΈ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 3041 | . . . 4 β’ ({π} β πΈ β Β¬ {π} β πΈ) | |
2 | clwwlknon1.v | . . . . . . . 8 β’ π = (VtxβπΊ) | |
3 | clwwlknon1.c | . . . . . . . 8 β’ πΆ = (ClWWalksNOnβπΊ) | |
4 | clwwlknon1.e | . . . . . . . 8 β’ πΈ = (EdgβπΊ) | |
5 | 2, 3, 4 | clwwlknon1nloop 29861 | . . . . . . 7 β’ ({π} β πΈ β (ππΆ1) = β ) |
6 | 5 | adantl 481 | . . . . . 6 β’ ((π β π β§ {π} β πΈ) β (ππΆ1) = β ) |
7 | s1cli 14561 | . . . . . . . . . 10 β’ β¨βπββ© β Word V | |
8 | 7 | elexi 3488 | . . . . . . . . 9 β’ β¨βπββ© β V |
9 | 8 | snnz 4775 | . . . . . . . 8 β’ {β¨βπββ©} β β |
10 | 9 | nesymi 2992 | . . . . . . 7 β’ Β¬ β = {β¨βπββ©} |
11 | eqeq1 2730 | . . . . . . 7 β’ ((ππΆ1) = β β ((ππΆ1) = {β¨βπββ©} β β = {β¨βπββ©})) | |
12 | 10, 11 | mtbiri 327 | . . . . . 6 β’ ((ππΆ1) = β β Β¬ (ππΆ1) = {β¨βπββ©}) |
13 | 6, 12 | syl 17 | . . . . 5 β’ ((π β π β§ {π} β πΈ) β Β¬ (ππΆ1) = {β¨βπββ©}) |
14 | 13 | ex 412 | . . . 4 β’ (π β π β ({π} β πΈ β Β¬ (ππΆ1) = {β¨βπββ©})) |
15 | 1, 14 | biimtrrid 242 | . . 3 β’ (π β π β (Β¬ {π} β πΈ β Β¬ (ππΆ1) = {β¨βπββ©})) |
16 | 15 | con4d 115 | . 2 β’ (π β π β ((ππΆ1) = {β¨βπββ©} β {π} β πΈ)) |
17 | 2, 3, 4 | clwwlknon1loop 29860 | . . 3 β’ ((π β π β§ {π} β πΈ) β (ππΆ1) = {β¨βπββ©}) |
18 | 17 | ex 412 | . 2 β’ (π β π β ({π} β πΈ β (ππΆ1) = {β¨βπββ©})) |
19 | 16, 18 | impbid 211 | 1 β’ (π β π β ((ππΆ1) = {β¨βπββ©} β {π} β πΈ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 β wnel 3040 Vcvv 3468 β c0 4317 {csn 4623 βcfv 6537 (class class class)co 7405 1c1 11113 Word cword 14470 β¨βcs1 14551 Vtxcvtx 28764 Edgcedg 28815 ClWWalksNOncclwwlknon 29849 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-lsw 14519 df-s1 14552 df-clwwlk 29744 df-clwwlkn 29787 df-clwwlknon 29850 |
This theorem is referenced by: (None) |
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