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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 29519 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ USPGraph) | 
| 6 | uspgrupgr 29196 | . . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
| 7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ UPGraph) | 
| 8 | 1 | eleq2d 2826 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ (Vtx‘𝐺) ↔ 𝑁 ∈ 𝑉)) | 
| 9 | 3, 8 | mpbird 257 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (Vtx‘𝐺)) | 
| 10 | eqid 2736 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 11 | eqid 2736 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
| 12 | 10, 11 | nbupgr 29362 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝑁) = {𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ {𝑁, 𝑣} ∈ (Edg‘𝐺)}) | 
| 13 | 7, 9, 12 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐺 NeighbVtx 𝑁) = {𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ {𝑁, 𝑣} ∈ (Edg‘𝐺)}) | 
| 14 | 1 | difeq1d 4124 | . . . . . . . 8 ⊢ (𝜑 → ((Vtx‘𝐺) ∖ {𝑁}) = (𝑉 ∖ {𝑁})) | 
| 15 | 14 | eleq2d 2826 | . . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ↔ 𝑣 ∈ (𝑉 ∖ {𝑁}))) | 
| 16 | eldifsn 4785 | . . . . . . . 8 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁)) | |
| 17 | 3 | adantr 480 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑁 ∈ 𝑉) | 
| 18 | simpr 484 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) | |
| 19 | 17, 18 | preqsnd 4858 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ({𝑁, 𝑣} = {𝑁} ↔ (𝑁 = 𝑁 ∧ 𝑣 = 𝑁))) | 
| 20 | simpr 484 | . . . . . . . . . . 11 ⊢ ((𝑁 = 𝑁 ∧ 𝑣 = 𝑁) → 𝑣 = 𝑁) | |
| 21 | 19, 20 | biimtrdi 253 | . . . . . . . . . 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 453 | . . . . . . . 8 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁) → ¬ {𝑁, 𝑣} = {𝑁})) | 
| 24 | 16, 23 | biimtrid 242 | . . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ {𝑁, 𝑣} = {𝑁})) | 
| 25 | 15, 24 | sylbid 240 | . . . . . 6 ⊢ (𝜑 → (𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) → ¬ {𝑁, 𝑣} = {𝑁})) | 
| 26 | 25 | imp 406 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ¬ {𝑁, 𝑣} = {𝑁}) | 
| 27 | 1, 2, 3, 4 | 1loopgredg 29520 | . . . . . . . . 9 ⊢ (𝜑 → (Edg‘𝐺) = {{𝑁}}) | 
| 28 | 27 | eleq2d 2826 | . . . . . . . 8 ⊢ (𝜑 → ({𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑁, 𝑣} ∈ {{𝑁}})) | 
| 29 | prex 5436 | . . . . . . . . 9 ⊢ {𝑁, 𝑣} ∈ V | |
| 30 | 29 | elsn 4640 | . . . . . . . 8 ⊢ ({𝑁, 𝑣} ∈ {{𝑁}} ↔ {𝑁, 𝑣} = {𝑁}) | 
| 31 | 28, 30 | bitrdi 287 | . . . . . . 7 ⊢ (𝜑 → ({𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑁, 𝑣} = {𝑁})) | 
| 32 | 31 | notbid 318 | . . . . . 6 ⊢ (𝜑 → (¬ {𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ ¬ {𝑁, 𝑣} = {𝑁})) | 
| 33 | 32 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → (¬ {𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ ¬ {𝑁, 𝑣} = {𝑁})) | 
| 34 | 26, 33 | mpbird 257 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ¬ {𝑁, 𝑣} ∈ (Edg‘𝐺)) | 
| 35 | 34 | ralrimiva 3145 | . . 3 ⊢ (𝜑 → ∀𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ {𝑁, 𝑣} ∈ (Edg‘𝐺)) | 
| 36 | rabeq0 4387 | . . 3 ⊢ ({𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ {𝑁, 𝑣} ∈ (Edg‘𝐺)} = ∅ ↔ ∀𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ {𝑁, 𝑣} ∈ (Edg‘𝐺)) | |
| 37 | 35, 36 | sylibr 234 | . 2 ⊢ (𝜑 → {𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ {𝑁, 𝑣} ∈ (Edg‘𝐺)} = ∅) | 
| 38 | 13, 37 | eqtrd 2776 | 1 ⊢ (𝜑 → (𝐺 NeighbVtx 𝑁) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 {crab 3435 ∖ cdif 3947 ∅c0 4332 {csn 4625 {cpr 4627 〈cop 4631 ‘cfv 6560 (class class class)co 7432 Vtxcvtx 29014 iEdgciedg 29015 Edgcedg 29065 UPGraphcupgr 29098 USPGraphcuspgr 29166 NeighbVtx cnbgr 29350 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-dju 9942 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12880 df-fz 13549 df-hash 14371 df-edg 29066 df-upgr 29100 df-uspgr 29168 df-nbgr 29351 | 
| This theorem is referenced by: uspgrloopnb0 29538 | 
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