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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 29356 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ USPGraph) |
6 | uspgrupgr 29033 | . . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ UPGraph) |
8 | 1 | eleq2d 2811 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ (Vtx‘𝐺) ↔ 𝑁 ∈ 𝑉)) |
9 | 3, 8 | mpbird 256 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (Vtx‘𝐺)) |
10 | eqid 2725 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
11 | eqid 2725 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
12 | 10, 11 | nbupgr 29199 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝑁) = {𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ {𝑁, 𝑣} ∈ (Edg‘𝐺)}) |
13 | 7, 9, 12 | syl2anc 582 | . 2 ⊢ (𝜑 → (𝐺 NeighbVtx 𝑁) = {𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ {𝑁, 𝑣} ∈ (Edg‘𝐺)}) |
14 | 1 | difeq1d 4113 | . . . . . . . 8 ⊢ (𝜑 → ((Vtx‘𝐺) ∖ {𝑁}) = (𝑉 ∖ {𝑁})) |
15 | 14 | eleq2d 2811 | . . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ↔ 𝑣 ∈ (𝑉 ∖ {𝑁}))) |
16 | eldifsn 4786 | . . . . . . . 8 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁)) | |
17 | 3 | adantr 479 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑁 ∈ 𝑉) |
18 | simpr 483 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) | |
19 | 17, 18 | preqsnd 4855 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ({𝑁, 𝑣} = {𝑁} ↔ (𝑁 = 𝑁 ∧ 𝑣 = 𝑁))) |
20 | simpr 483 | . . . . . . . . . . 11 ⊢ ((𝑁 = 𝑁 ∧ 𝑣 = 𝑁) → 𝑣 = 𝑁) | |
21 | 19, 20 | biimtrdi 252 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ({𝑁, 𝑣} = {𝑁} → 𝑣 = 𝑁)) |
22 | 21 | necon3ad 2943 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑣 ≠ 𝑁 → ¬ {𝑁, 𝑣} = {𝑁})) |
23 | 22 | expimpd 452 | . . . . . . . 8 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁) → ¬ {𝑁, 𝑣} = {𝑁})) |
24 | 16, 23 | biimtrid 241 | . . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ {𝑁, 𝑣} = {𝑁})) |
25 | 15, 24 | sylbid 239 | . . . . . 6 ⊢ (𝜑 → (𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) → ¬ {𝑁, 𝑣} = {𝑁})) |
26 | 25 | imp 405 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ¬ {𝑁, 𝑣} = {𝑁}) |
27 | 1, 2, 3, 4 | 1loopgredg 29357 | . . . . . . . . 9 ⊢ (𝜑 → (Edg‘𝐺) = {{𝑁}}) |
28 | 27 | eleq2d 2811 | . . . . . . . 8 ⊢ (𝜑 → ({𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑁, 𝑣} ∈ {{𝑁}})) |
29 | prex 5428 | . . . . . . . . 9 ⊢ {𝑁, 𝑣} ∈ V | |
30 | 29 | elsn 4639 | . . . . . . . 8 ⊢ ({𝑁, 𝑣} ∈ {{𝑁}} ↔ {𝑁, 𝑣} = {𝑁}) |
31 | 28, 30 | bitrdi 286 | . . . . . . 7 ⊢ (𝜑 → ({𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑁, 𝑣} = {𝑁})) |
32 | 31 | notbid 317 | . . . . . 6 ⊢ (𝜑 → (¬ {𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ ¬ {𝑁, 𝑣} = {𝑁})) |
33 | 32 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → (¬ {𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ ¬ {𝑁, 𝑣} = {𝑁})) |
34 | 26, 33 | mpbird 256 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ¬ {𝑁, 𝑣} ∈ (Edg‘𝐺)) |
35 | 34 | ralrimiva 3136 | . . 3 ⊢ (𝜑 → ∀𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ {𝑁, 𝑣} ∈ (Edg‘𝐺)) |
36 | rabeq0 4380 | . . 3 ⊢ ({𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ {𝑁, 𝑣} ∈ (Edg‘𝐺)} = ∅ ↔ ∀𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ {𝑁, 𝑣} ∈ (Edg‘𝐺)) | |
37 | 35, 36 | sylibr 233 | . 2 ⊢ (𝜑 → {𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ {𝑁, 𝑣} ∈ (Edg‘𝐺)} = ∅) |
38 | 13, 37 | eqtrd 2765 | 1 ⊢ (𝜑 → (𝐺 NeighbVtx 𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 {crab 3419 ∖ cdif 3937 ∅c0 4318 {csn 4624 {cpr 4626 ⟨cop 4630 ‘cfv 6542 (class class class)co 7415 Vtxcvtx 28851 iEdgciedg 28852 Edgcedg 28902 UPGraphcupgr 28935 USPGraphcuspgr 29003 NeighbVtx cnbgr 29187 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-oadd 8487 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-dju 9922 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-n0 12501 df-xnn0 12573 df-z 12587 df-uz 12851 df-fz 13515 df-hash 14320 df-edg 28903 df-upgr 28937 df-uspgr 29005 df-nbgr 29188 |
This theorem is referenced by: uspgrloopnb0 29375 |
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