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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 6557 | . . . 4 ⊢ 𝑉 ∈ V |
3 | difexg 5127 | . . . 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 1189 | . . . 4 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈)) | |
6 | 1 | nbgrssvwo2 26832 | . . . 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 13623 | . . 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 26789 | . . . 4 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin) |
11 | 10 | ad2antrr 722 | . . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → 𝑉 ∈ Fin) |
12 | simpr1 1187 | . . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → 𝑀 ∈ 𝑉) | |
13 | simplr 765 | . . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → 𝑈 ∈ 𝑉) | |
14 | simpr2 1188 | . . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → 𝑀 ≠ 𝑈) | |
15 | hashdifpr 13629 | . . 3 ⊢ ((𝑉 ∈ Fin ∧ (𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → (♯‘(𝑉 ∖ {𝑀, 𝑈})) = ((♯‘𝑉) − 2)) | |
16 | 11, 12, 13, 14, 15 | syl13anc 1365 | . 2 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → (♯‘(𝑉 ∖ {𝑀, 𝑈})) = ((♯‘𝑉) − 2)) |
17 | 9, 16 | breqtrd 4992 | 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 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∉ wnel 3090 Vcvv 3437 ∖ cdif 3860 ⊆ wss 3863 {cpr 4478 class class class wbr 4966 ‘cfv 6230 (class class class)co 7021 Fincfn 8362 ≤ cle 10527 − cmin 10722 2c2 11545 ♯chash 13545 Vtxcvtx 26469 FinUSGraphcfusgr 26786 NeighbVtx cnbgr 26802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-oadd 7962 df-er 8144 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-dju 9181 df-card 9219 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-2 11553 df-n0 11751 df-xnn0 11821 df-z 11835 df-uz 12099 df-fz 12748 df-hash 13546 df-fusgr 26787 df-nbgr 26803 |
This theorem is referenced by: nbusgrvtxm1 26849 |
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