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Mirrors > Home > MPE Home > Th. List > clwwlk0on0 | Structured version Visualization version GIF version |
Description: There is no word over the set of vertices representing a closed walk on vertex π of length 0 in a graph πΊ. (Contributed by AV, 17-Feb-2022.) (Revised by AV, 25-Feb-2022.) |
Ref | Expression |
---|---|
clwwlk0on0 | β’ (π(ClWWalksNOnβπΊ)0) = β |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2748 | . . . . 5 β’ (π£ = π β ((π€β0) = π£ β (π€β0) = π)) | |
2 | 1 | rabbidv 3415 | . . . 4 β’ (π£ = π β {π€ β (π ClWWalksN πΊ) β£ (π€β0) = π£} = {π€ β (π ClWWalksN πΊ) β£ (π€β0) = π}) |
3 | oveq1 7364 | . . . . . 6 β’ (π = 0 β (π ClWWalksN πΊ) = (0 ClWWalksN πΊ)) | |
4 | clwwlkn0 28972 | . . . . . 6 β’ (0 ClWWalksN πΊ) = β | |
5 | 3, 4 | eqtrdi 2792 | . . . . 5 β’ (π = 0 β (π ClWWalksN πΊ) = β ) |
6 | 5 | rabeqdv 3422 | . . . 4 β’ (π = 0 β {π€ β (π ClWWalksN πΊ) β£ (π€β0) = π} = {π€ β β β£ (π€β0) = π}) |
7 | clwwlknonmpo 29033 | . . . 4 β’ (ClWWalksNOnβπΊ) = (π£ β (VtxβπΊ), π β β0 β¦ {π€ β (π ClWWalksN πΊ) β£ (π€β0) = π£}) | |
8 | 0ex 5264 | . . . . 5 β’ β β V | |
9 | 8 | rabex 5289 | . . . 4 β’ {π€ β β β£ (π€β0) = π} β V |
10 | 2, 6, 7, 9 | ovmpo 7515 | . . 3 β’ ((π β (VtxβπΊ) β§ 0 β β0) β (π(ClWWalksNOnβπΊ)0) = {π€ β β β£ (π€β0) = π}) |
11 | rab0 4342 | . . 3 β’ {π€ β β β£ (π€β0) = π} = β | |
12 | 10, 11 | eqtrdi 2792 | . 2 β’ ((π β (VtxβπΊ) β§ 0 β β0) β (π(ClWWalksNOnβπΊ)0) = β ) |
13 | 7 | mpondm0 7594 | . 2 β’ (Β¬ (π β (VtxβπΊ) β§ 0 β β0) β (π(ClWWalksNOnβπΊ)0) = β ) |
14 | 12, 13 | pm2.61i 182 | 1 β’ (π(ClWWalksNOnβπΊ)0) = β |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 396 = wceq 1541 β wcel 2106 {crab 3407 β c0 4282 βcfv 6496 (class class class)co 7357 0cc0 11051 β0cn0 12413 Vtxcvtx 27947 ClWWalksN cclwwlkn 28968 ClWWalksNOncclwwlknon 29031 |
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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568 df-hash 14231 df-word 14403 df-clwwlk 28926 df-clwwlkn 28969 df-clwwlknon 29032 |
This theorem is referenced by: clwwlknon0 29037 |
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