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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 29533 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ USPGraph) |
6 | uspgrupgr 29210 | . . . 4 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ UPGraph) |
8 | 1 | eleq2d 2825 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ (Vtx‘𝐺) ↔ 𝑁 ∈ 𝑉)) |
9 | 3, 8 | mpbird 257 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (Vtx‘𝐺)) |
10 | eqid 2735 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
11 | eqid 2735 | . . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
12 | 10, 11 | nbupgr 29376 | . . 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 4135 | . . . . . . . 8 ⊢ (𝜑 → ((Vtx‘𝐺) ∖ {𝑁}) = (𝑉 ∖ {𝑁})) |
15 | 14 | eleq2d 2825 | . . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ↔ 𝑣 ∈ (𝑉 ∖ {𝑁}))) |
16 | eldifsn 4791 | . . . . . . . 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 4864 | . . . . . . . . . . 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 2951 | . . . . . . . . 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 29534 | . . . . . . . . 9 ⊢ (𝜑 → (Edg‘𝐺) = {{𝑁}}) |
28 | 27 | eleq2d 2825 | . . . . . . . 8 ⊢ (𝜑 → ({𝑁, 𝑣} ∈ (Edg‘𝐺) ↔ {𝑁, 𝑣} ∈ {{𝑁}})) |
29 | prex 5443 | . . . . . . . . 9 ⊢ {𝑁, 𝑣} ∈ V | |
30 | 29 | elsn 4646 | . . . . . . . 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 3144 | . . 3 ⊢ (𝜑 → ∀𝑣 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ {𝑁, 𝑣} ∈ (Edg‘𝐺)) |
36 | rabeq0 4394 | . . 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 2775 | 1 ⊢ (𝜑 → (𝐺 NeighbVtx 𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 {crab 3433 ∖ cdif 3960 ∅c0 4339 {csn 4631 {cpr 4633 〈cop 4637 ‘cfv 6563 (class class class)co 7431 Vtxcvtx 29028 iEdgciedg 29029 Edgcedg 29079 UPGraphcupgr 29112 USPGraphcuspgr 29180 NeighbVtx cnbgr 29364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 df-edg 29080 df-upgr 29114 df-uspgr 29182 df-nbgr 29365 |
This theorem is referenced by: uspgrloopnb0 29552 |
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