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Mirrors > Home > MPE Home > Th. List > 1loopgrnb0 | Structured version Visualization version GIF version |
Description: In a graph (simple pseudograph) with one edge which is a loop, the vertex connected with itself by the loop has no neighbors. (Contributed by AV, 17-Dec-2020.) (Revised by AV, 21-Feb-2021.) |
Ref | Expression |
---|---|
1loopgruspgr.v | ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) |
1loopgruspgr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
1loopgruspgr.n | ⊢ (𝜑 → 𝑁 ∈ 𝑉) |
1loopgruspgr.i | ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝑁}〉}) |
Ref | Expression |
---|---|
1loopgrnb0 | ⊢ (𝜑 → (𝐺 NeighbVtx 𝑁) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1loopgruspgr.v | . . . . 5 ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) | |
2 | 1loopgruspgr.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
3 | 1loopgruspgr.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ 𝑉) | |
4 | 1loopgruspgr.i | . . . . 5 ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝑁}〉}) | |
5 | 1, 2, 3, 4 | 1loopgruspgr 27585 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ USPGraph) |
6 | uspgrupgr 27264 | . . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ UPGraph) |
8 | 1 | eleq2d 2823 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ (Vtx‘𝐺) ↔ 𝑁 ∈ 𝑉)) |
9 | 3, 8 | mpbird 260 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (Vtx‘𝐺)) |
10 | eqid 2737 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
11 | eqid 2737 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
12 | 10, 11 | nbupgr 27429 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝑁) = {𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ {𝑁, 𝑣} ∈ (Edg‘𝐺)}) |
13 | 7, 9, 12 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐺 NeighbVtx 𝑁) = {𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ {𝑁, 𝑣} ∈ (Edg‘𝐺)}) |
14 | 1 | difeq1d 4033 | . . . . . . . 8 ⊢ (𝜑 → ((Vtx‘𝐺) ∖ {𝑁}) = (𝑉 ∖ {𝑁})) |
15 | 14 | eleq2d 2823 | . . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ↔ 𝑣 ∈ (𝑉 ∖ {𝑁}))) |
16 | eldifsn 4697 | . . . . . . . 8 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁)) | |
17 | 3 | adantr 484 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑁 ∈ 𝑉) |
18 | simpr 488 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) | |
19 | 17, 18 | preqsnd 4766 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ({𝑁, 𝑣} = {𝑁} ↔ (𝑁 = 𝑁 ∧ 𝑣 = 𝑁))) |
20 | simpr 488 | . . . . . . . . . . 11 ⊢ ((𝑁 = 𝑁 ∧ 𝑣 = 𝑁) → 𝑣 = 𝑁) | |
21 | 19, 20 | syl6bi 256 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ({𝑁, 𝑣} = {𝑁} → 𝑣 = 𝑁)) |
22 | 21 | necon3ad 2952 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑣 ≠ 𝑁 → ¬ {𝑁, 𝑣} = {𝑁})) |
23 | 22 | expimpd 457 | . . . . . . . 8 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁) → ¬ {𝑁, 𝑣} = {𝑁})) |
24 | 16, 23 | syl5bi 245 | . . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ {𝑁, 𝑣} = {𝑁})) |
25 | 15, 24 | sylbid 243 | . . . . . 6 ⊢ (𝜑 → (𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) → ¬ {𝑁, 𝑣} = {𝑁})) |
26 | 25 | imp 410 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ¬ {𝑁, 𝑣} = {𝑁}) |
27 | 1, 2, 3, 4 | 1loopgredg 27586 | . . . . . . . . 9 ⊢ (𝜑 → (Edg‘𝐺) = {{𝑁}}) |
28 | 27 | eleq2d 2823 | . . . . . . . 8 ⊢ (𝜑 → ({𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑁, 𝑣} ∈ {{𝑁}})) |
29 | prex 5322 | . . . . . . . . 9 ⊢ {𝑁, 𝑣} ∈ V | |
30 | 29 | elsn 4553 | . . . . . . . 8 ⊢ ({𝑁, 𝑣} ∈ {{𝑁}} ↔ {𝑁, 𝑣} = {𝑁}) |
31 | 28, 30 | bitrdi 290 | . . . . . . 7 ⊢ (𝜑 → ({𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑁, 𝑣} = {𝑁})) |
32 | 31 | notbid 321 | . . . . . 6 ⊢ (𝜑 → (¬ {𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ ¬ {𝑁, 𝑣} = {𝑁})) |
33 | 32 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → (¬ {𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ ¬ {𝑁, 𝑣} = {𝑁})) |
34 | 26, 33 | mpbird 260 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ¬ {𝑁, 𝑣} ∈ (Edg‘𝐺)) |
35 | 34 | ralrimiva 3102 | . . 3 ⊢ (𝜑 → ∀𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ {𝑁, 𝑣} ∈ (Edg‘𝐺)) |
36 | rabeq0 4296 | . . 3 ⊢ ({𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ {𝑁, 𝑣} ∈ (Edg‘𝐺)} = ∅ ↔ ∀𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ {𝑁, 𝑣} ∈ (Edg‘𝐺)) | |
37 | 35, 36 | sylibr 237 | . 2 ⊢ (𝜑 → {𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ {𝑁, 𝑣} ∈ (Edg‘𝐺)} = ∅) |
38 | 13, 37 | eqtrd 2777 | 1 ⊢ (𝜑 → (𝐺 NeighbVtx 𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2939 ∀wral 3058 {crab 3062 ∖ cdif 3860 ∅c0 4234 {csn 4538 {cpr 4540 〈cop 4544 ‘cfv 6377 (class class class)co 7210 Vtxcvtx 27084 iEdgciedg 27085 Edgcedg 27135 UPGraphcupgr 27168 USPGraphcuspgr 27236 NeighbVtx cnbgr 27417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-int 4857 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-1st 7758 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-1o 8199 df-2o 8200 df-oadd 8203 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-fin 8627 df-dju 9514 df-card 9552 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-nn 11828 df-2 11890 df-n0 12088 df-xnn0 12160 df-z 12174 df-uz 12436 df-fz 13093 df-hash 13894 df-edg 27136 df-upgr 27170 df-uspgr 27238 df-nbgr 27418 |
This theorem is referenced by: uspgrloopnb0 27604 |
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