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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 27000 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ USPGraph) |
6 | uspgrupgr 26679 | . . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ UPGraph) |
8 | 1 | eleq2d 2844 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ (Vtx‘𝐺) ↔ 𝑁 ∈ 𝑉)) |
9 | 3, 8 | mpbird 249 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (Vtx‘𝐺)) |
10 | eqid 2771 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
11 | eqid 2771 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
12 | 10, 11 | nbupgr 26844 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝑁) = {𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ {𝑁, 𝑣} ∈ (Edg‘𝐺)}) |
13 | 7, 9, 12 | syl2anc 576 | . 2 ⊢ (𝜑 → (𝐺 NeighbVtx 𝑁) = {𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ {𝑁, 𝑣} ∈ (Edg‘𝐺)}) |
14 | 1 | difeq1d 3981 | . . . . . . . 8 ⊢ (𝜑 → ((Vtx‘𝐺) ∖ {𝑁}) = (𝑉 ∖ {𝑁})) |
15 | 14 | eleq2d 2844 | . . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ↔ 𝑣 ∈ (𝑉 ∖ {𝑁}))) |
16 | eldifsn 4589 | . . . . . . . 8 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁)) | |
17 | 3 | elexd 3428 | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ V) |
18 | 17 | adantr 473 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑁 ∈ V) |
19 | elex 3426 | . . . . . . . . . . . . 13 ⊢ (𝑣 ∈ 𝑉 → 𝑣 ∈ V) | |
20 | 19 | adantl 474 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ V) |
21 | 18, 20 | preqsnd 4659 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ({𝑁, 𝑣} = {𝑁} ↔ (𝑁 = 𝑁 ∧ 𝑣 = 𝑁))) |
22 | simpr 477 | . . . . . . . . . . 11 ⊢ ((𝑁 = 𝑁 ∧ 𝑣 = 𝑁) → 𝑣 = 𝑁) | |
23 | 21, 22 | syl6bi 245 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ({𝑁, 𝑣} = {𝑁} → 𝑣 = 𝑁)) |
24 | 23 | necon3ad 2973 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑣 ≠ 𝑁 → ¬ {𝑁, 𝑣} = {𝑁})) |
25 | 24 | expimpd 446 | . . . . . . . 8 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁) → ¬ {𝑁, 𝑣} = {𝑁})) |
26 | 16, 25 | syl5bi 234 | . . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ {𝑁, 𝑣} = {𝑁})) |
27 | 15, 26 | sylbid 232 | . . . . . 6 ⊢ (𝜑 → (𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) → ¬ {𝑁, 𝑣} = {𝑁})) |
28 | 27 | imp 398 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ¬ {𝑁, 𝑣} = {𝑁}) |
29 | 1, 2, 3, 4 | 1loopgredg 27001 | . . . . . . . . 9 ⊢ (𝜑 → (Edg‘𝐺) = {{𝑁}}) |
30 | 29 | eleq2d 2844 | . . . . . . . 8 ⊢ (𝜑 → ({𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑁, 𝑣} ∈ {{𝑁}})) |
31 | prex 5185 | . . . . . . . . 9 ⊢ {𝑁, 𝑣} ∈ V | |
32 | 31 | elsn 4450 | . . . . . . . 8 ⊢ ({𝑁, 𝑣} ∈ {{𝑁}} ↔ {𝑁, 𝑣} = {𝑁}) |
33 | 30, 32 | syl6bb 279 | . . . . . . 7 ⊢ (𝜑 → ({𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑁, 𝑣} = {𝑁})) |
34 | 33 | notbid 310 | . . . . . 6 ⊢ (𝜑 → (¬ {𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ ¬ {𝑁, 𝑣} = {𝑁})) |
35 | 34 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → (¬ {𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ ¬ {𝑁, 𝑣} = {𝑁})) |
36 | 28, 35 | mpbird 249 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ¬ {𝑁, 𝑣} ∈ (Edg‘𝐺)) |
37 | 36 | ralrimiva 3125 | . . 3 ⊢ (𝜑 → ∀𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ {𝑁, 𝑣} ∈ (Edg‘𝐺)) |
38 | rabeq0 4218 | . . 3 ⊢ ({𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ {𝑁, 𝑣} ∈ (Edg‘𝐺)} = ∅ ↔ ∀𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ {𝑁, 𝑣} ∈ (Edg‘𝐺)) | |
39 | 37, 38 | sylibr 226 | . 2 ⊢ (𝜑 → {𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ {𝑁, 𝑣} ∈ (Edg‘𝐺)} = ∅) |
40 | 13, 39 | eqtrd 2807 | 1 ⊢ (𝜑 → (𝐺 NeighbVtx 𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ≠ wne 2960 ∀wral 3081 {crab 3085 Vcvv 3408 ∖ cdif 3819 ∅c0 4172 {csn 4435 {cpr 4437 〈cop 4441 ‘cfv 6185 (class class class)co 6974 Vtxcvtx 26499 iEdgciedg 26500 Edgcedg 26550 UPGraphcupgr 26583 USPGraphcuspgr 26651 NeighbVtx cnbgr 26832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-dju 9122 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-n0 11706 df-xnn0 11778 df-z 11792 df-uz 12057 df-fz 12707 df-hash 13504 df-edg 26551 df-upgr 26585 df-uspgr 26653 df-nbgr 26833 |
This theorem is referenced by: uspgrloopnb0 27019 |
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