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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 𝑋 of length 1 as words over the set of vertices is a singleton containing the singleton word consisting of 𝑋 iff there is a loop at 𝑋. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.) |
Ref | Expression |
---|---|
clwwlknon1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
clwwlknon1.c | ⊢ 𝐶 = (ClWWalksNOn‘𝐺) |
clwwlknon1.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
clwwlknon1sn | ⊢ (𝑋 ∈ 𝑉 → ((𝑋𝐶1) = {〈“𝑋”〉} ↔ {𝑋} ∈ 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 3124 | . . . 4 ⊢ ({𝑋} ∉ 𝐸 ↔ ¬ {𝑋} ∈ 𝐸) | |
2 | clwwlknon1.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | clwwlknon1.c | . . . . . . . 8 ⊢ 𝐶 = (ClWWalksNOn‘𝐺) | |
4 | clwwlknon1.e | . . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺) | |
5 | 2, 3, 4 | clwwlknon1nloop 27862 | . . . . . . 7 ⊢ ({𝑋} ∉ 𝐸 → (𝑋𝐶1) = ∅) |
6 | 5 | adantl 484 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∉ 𝐸) → (𝑋𝐶1) = ∅) |
7 | s1cli 13944 | . . . . . . . . . 10 ⊢ 〈“𝑋”〉 ∈ Word V | |
8 | 7 | elexi 3505 | . . . . . . . . 9 ⊢ 〈“𝑋”〉 ∈ V |
9 | 8 | snnz 4697 | . . . . . . . 8 ⊢ {〈“𝑋”〉} ≠ ∅ |
10 | 9 | nesymi 3073 | . . . . . . 7 ⊢ ¬ ∅ = {〈“𝑋”〉} |
11 | eqeq1 2825 | . . . . . . 7 ⊢ ((𝑋𝐶1) = ∅ → ((𝑋𝐶1) = {〈“𝑋”〉} ↔ ∅ = {〈“𝑋”〉})) | |
12 | 10, 11 | mtbiri 329 | . . . . . 6 ⊢ ((𝑋𝐶1) = ∅ → ¬ (𝑋𝐶1) = {〈“𝑋”〉}) |
13 | 6, 12 | syl 17 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∉ 𝐸) → ¬ (𝑋𝐶1) = {〈“𝑋”〉}) |
14 | 13 | ex 415 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → ({𝑋} ∉ 𝐸 → ¬ (𝑋𝐶1) = {〈“𝑋”〉})) |
15 | 1, 14 | syl5bir 245 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (¬ {𝑋} ∈ 𝐸 → ¬ (𝑋𝐶1) = {〈“𝑋”〉})) |
16 | 15 | con4d 115 | . 2 ⊢ (𝑋 ∈ 𝑉 → ((𝑋𝐶1) = {〈“𝑋”〉} → {𝑋} ∈ 𝐸)) |
17 | 2, 3, 4 | clwwlknon1loop 27861 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → (𝑋𝐶1) = {〈“𝑋”〉}) |
18 | 17 | ex 415 | . 2 ⊢ (𝑋 ∈ 𝑉 → ({𝑋} ∈ 𝐸 → (𝑋𝐶1) = {〈“𝑋”〉})) |
19 | 16, 18 | impbid 214 | 1 ⊢ (𝑋 ∈ 𝑉 → ((𝑋𝐶1) = {〈“𝑋”〉} ↔ {𝑋} ∈ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∉ wnel 3123 Vcvv 3486 ∅c0 4279 {csn 4553 ‘cfv 6341 (class class class)co 7142 1c1 10524 Word cword 13851 〈“cs1 13934 Vtxcvtx 26767 Edgcedg 26818 ClWWalksNOncclwwlknon 27850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-map 8394 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-n0 11885 df-xnn0 11955 df-z 11969 df-uz 12231 df-fz 12883 df-fzo 13024 df-hash 13681 df-word 13852 df-lsw 13900 df-s1 13935 df-clwwlk 27745 df-clwwlkn 27788 df-clwwlknon 27851 |
This theorem is referenced by: (None) |
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