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Mirrors > Home > MPE Home > Th. List > clwwlknon1nloop | Structured version Visualization version GIF version |
Description: If there is no loop at vertex 𝑋, the set of (closed) walks on 𝑋 of length 1 as words over the set of vertices is empty. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Mar-2022.) |
Ref | Expression |
---|---|
clwwlknon1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
clwwlknon1.c | ⊢ 𝐶 = (ClWWalksNOn‘𝐺) |
clwwlknon1.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
clwwlknon1nloop | ⊢ ({𝑋} ∉ 𝐸 → (𝑋𝐶1) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwwlknon1.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | clwwlknon1.c | . . . . 5 ⊢ 𝐶 = (ClWWalksNOn‘𝐺) | |
3 | clwwlknon1.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
4 | 1, 2, 3 | clwwlknon1 27878 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)}) |
5 | 4 | adantr 483 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∉ 𝐸) → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)}) |
6 | df-nel 3126 | . . . . . . . . 9 ⊢ ({𝑋} ∉ 𝐸 ↔ ¬ {𝑋} ∈ 𝐸) | |
7 | 6 | biimpi 218 | . . . . . . . 8 ⊢ ({𝑋} ∉ 𝐸 → ¬ {𝑋} ∈ 𝐸) |
8 | 7 | olcd 870 | . . . . . . 7 ⊢ ({𝑋} ∉ 𝐸 → (¬ 𝑤 = 〈“𝑋”〉 ∨ ¬ {𝑋} ∈ 𝐸)) |
9 | 8 | ad2antlr 725 | . . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∉ 𝐸) ∧ 𝑤 ∈ Word 𝑉) → (¬ 𝑤 = 〈“𝑋”〉 ∨ ¬ {𝑋} ∈ 𝐸)) |
10 | ianor 978 | . . . . . 6 ⊢ (¬ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸) ↔ (¬ 𝑤 = 〈“𝑋”〉 ∨ ¬ {𝑋} ∈ 𝐸)) | |
11 | 9, 10 | sylibr 236 | . . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∉ 𝐸) ∧ 𝑤 ∈ Word 𝑉) → ¬ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)) |
12 | 11 | ralrimiva 3184 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∉ 𝐸) → ∀𝑤 ∈ Word 𝑉 ¬ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)) |
13 | rabeq0 4340 | . . . 4 ⊢ ({𝑤 ∈ Word 𝑉 ∣ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)} = ∅ ↔ ∀𝑤 ∈ Word 𝑉 ¬ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)) | |
14 | 12, 13 | sylibr 236 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∉ 𝐸) → {𝑤 ∈ Word 𝑉 ∣ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)} = ∅) |
15 | 5, 14 | eqtrd 2858 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∉ 𝐸) → (𝑋𝐶1) = ∅) |
16 | 2 | oveqi 7171 | . . . 4 ⊢ (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1) |
17 | 1 | eleq2i 2906 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 ↔ 𝑋 ∈ (Vtx‘𝐺)) |
18 | 17 | notbii 322 | . . . . . . 7 ⊢ (¬ 𝑋 ∈ 𝑉 ↔ ¬ 𝑋 ∈ (Vtx‘𝐺)) |
19 | 18 | biimpi 218 | . . . . . 6 ⊢ (¬ 𝑋 ∈ 𝑉 → ¬ 𝑋 ∈ (Vtx‘𝐺)) |
20 | 19 | intnanrd 492 | . . . . 5 ⊢ (¬ 𝑋 ∈ 𝑉 → ¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 1 ∈ ℕ)) |
21 | clwwlknon0 27874 | . . . . 5 ⊢ (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 1 ∈ ℕ) → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) | |
22 | 20, 21 | syl 17 | . . . 4 ⊢ (¬ 𝑋 ∈ 𝑉 → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) |
23 | 16, 22 | syl5eq 2870 | . . 3 ⊢ (¬ 𝑋 ∈ 𝑉 → (𝑋𝐶1) = ∅) |
24 | 23 | adantr 483 | . 2 ⊢ ((¬ 𝑋 ∈ 𝑉 ∧ {𝑋} ∉ 𝐸) → (𝑋𝐶1) = ∅) |
25 | 15, 24 | pm2.61ian 810 | 1 ⊢ ({𝑋} ∉ 𝐸 → (𝑋𝐶1) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∉ wnel 3125 ∀wral 3140 {crab 3144 ∅c0 4293 {csn 4569 ‘cfv 6357 (class class class)co 7158 1c1 10540 ℕcn 11640 Word cword 13864 〈“cs1 13951 Vtxcvtx 26783 Edgcedg 26834 ClWWalksNOncclwwlknon 27868 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-lsw 13917 df-s1 13952 df-clwwlk 27762 df-clwwlkn 27805 df-clwwlknon 27869 |
This theorem is referenced by: clwwlknon1sn 27881 clwwlknon1le1 27882 |
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