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Mirrors > Home > MPE Home > Th. List > nbfusgrlevtxm2 | Structured version Visualization version GIF version |
Description: If there is a vertex which is not a neighbor of another vertex, the number of neighbors of the other vertex is at most the number of vertices of the graph minus 2 in a finite simple graph. (Contributed by AV, 16-Dec-2020.) (Proof shortened by AV, 13-Feb-2022.) |
Ref | Expression |
---|---|
hashnbusgrnn0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
nbfusgrlevtxm2 | ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → (♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashnbusgrnn0.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | fvexi 6853 | . . . 4 ⊢ 𝑉 ∈ V |
3 | difexg 5282 | . . . 4 ⊢ (𝑉 ∈ V → (𝑉 ∖ {𝑀, 𝑈}) ∈ V) | |
4 | 2, 3 | mp1i 13 | . . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → (𝑉 ∖ {𝑀, 𝑈}) ∈ V) |
5 | simpr3 1196 | . . . 4 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈)) | |
6 | 1 | nbgrssvwo2 28196 | . . . 4 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑈) → (𝐺 NeighbVtx 𝑈) ⊆ (𝑉 ∖ {𝑀, 𝑈})) |
7 | 5, 6 | syl 17 | . . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → (𝐺 NeighbVtx 𝑈) ⊆ (𝑉 ∖ {𝑀, 𝑈})) |
8 | hashss 14301 | . . 3 ⊢ (((𝑉 ∖ {𝑀, 𝑈}) ∈ V ∧ (𝐺 NeighbVtx 𝑈) ⊆ (𝑉 ∖ {𝑀, 𝑈})) → (♯‘(𝐺 NeighbVtx 𝑈)) ≤ (♯‘(𝑉 ∖ {𝑀, 𝑈}))) | |
9 | 4, 7, 8 | syl2anc 584 | . 2 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → (♯‘(𝐺 NeighbVtx 𝑈)) ≤ (♯‘(𝑉 ∖ {𝑀, 𝑈}))) |
10 | 1 | fusgrvtxfi 28153 | . . . 4 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin) |
11 | 10 | ad2antrr 724 | . . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → 𝑉 ∈ Fin) |
12 | simpr1 1194 | . . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → 𝑀 ∈ 𝑉) | |
13 | simplr 767 | . . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → 𝑈 ∈ 𝑉) | |
14 | simpr2 1195 | . . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → 𝑀 ≠ 𝑈) | |
15 | hashdifpr 14307 | . . 3 ⊢ ((𝑉 ∈ Fin ∧ (𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → (♯‘(𝑉 ∖ {𝑀, 𝑈})) = ((♯‘𝑉) − 2)) | |
16 | 11, 12, 13, 14, 15 | syl13anc 1372 | . 2 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → (♯‘(𝑉 ∖ {𝑀, 𝑈})) = ((♯‘𝑉) − 2)) |
17 | 9, 16 | breqtrd 5129 | 1 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → (♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∉ wnel 3047 Vcvv 3443 ∖ cdif 3905 ⊆ wss 3908 {cpr 4586 class class class wbr 5103 ‘cfv 6493 (class class class)co 7353 Fincfn 8879 ≤ cle 11186 − cmin 11381 2c2 12204 ♯chash 14222 Vtxcvtx 27833 FinUSGraphcfusgr 28150 NeighbVtx cnbgr 28166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-oadd 8412 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-dju 9833 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-n0 12410 df-xnn0 12482 df-z 12496 df-uz 12760 df-fz 13417 df-hash 14223 df-fusgr 28151 df-nbgr 28167 |
This theorem is referenced by: nbusgrvtxm1 28213 |
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