![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 𝑋 of length 0 in a graph 𝐺. (Contributed by AV, 17-Feb-2022.) (Revised by AV, 25-Feb-2022.) |
Ref | Expression |
---|---|
clwwlk0on0 | ⊢ (𝑋(ClWWalksNOn‘𝐺)0) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2808 | . . . . 5 ⊢ (𝑣 = 𝑋 → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑋)) | |
2 | 1 | rabbidv 3371 | . . . 4 ⊢ (𝑣 = 𝑋 → {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣} = {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) |
3 | oveq1 6883 | . . . . . 6 ⊢ (𝑛 = 0 → (𝑛 ClWWalksN 𝐺) = (0 ClWWalksN 𝐺)) | |
4 | clwwlkn0 27325 | . . . . . 6 ⊢ (0 ClWWalksN 𝐺) = ∅ | |
5 | 3, 4 | syl6eq 2847 | . . . . 5 ⊢ (𝑛 = 0 → (𝑛 ClWWalksN 𝐺) = ∅) |
6 | 5 | rabeqdv 3376 | . . . 4 ⊢ (𝑛 = 0 → {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ ∅ ∣ (𝑤‘0) = 𝑋}) |
7 | clwwlknonmpt2 27416 | . . . 4 ⊢ (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) | |
8 | 0ex 4982 | . . . . 5 ⊢ ∅ ∈ V | |
9 | 8 | rabex 5005 | . . . 4 ⊢ {𝑤 ∈ ∅ ∣ (𝑤‘0) = 𝑋} ∈ V |
10 | 2, 6, 7, 9 | ovmpt2 7028 | . . 3 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ 0 ∈ ℕ0) → (𝑋(ClWWalksNOn‘𝐺)0) = {𝑤 ∈ ∅ ∣ (𝑤‘0) = 𝑋}) |
11 | rab0 4154 | . . 3 ⊢ {𝑤 ∈ ∅ ∣ (𝑤‘0) = 𝑋} = ∅ | |
12 | 10, 11 | syl6eq 2847 | . 2 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ 0 ∈ ℕ0) → (𝑋(ClWWalksNOn‘𝐺)0) = ∅) |
13 | 7 | mpt2ndm0 7107 | . 2 ⊢ (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 0 ∈ ℕ0) → (𝑋(ClWWalksNOn‘𝐺)0) = ∅) |
14 | 12, 13 | pm2.61i 177 | 1 ⊢ (𝑋(ClWWalksNOn‘𝐺)0) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 385 = wceq 1653 ∈ wcel 2157 {crab 3091 ∅c0 4113 ‘cfv 6099 (class class class)co 6876 0cc0 10222 ℕ0cn0 11576 Vtxcvtx 26222 ClWWalksN cclwwlkn 27317 ClWWalksNOncclwwlknon 27414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-map 8095 df-pm 8096 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-card 9049 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-n0 11577 df-xnn0 11649 df-z 11663 df-uz 11927 df-fz 12577 df-fzo 12717 df-hash 13367 df-word 13531 df-clwwlk 27266 df-clwwlkn 27319 df-clwwlknon 27415 |
This theorem is referenced by: clwwlknon0 27422 |
Copyright terms: Public domain | W3C validator |