![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nbfusgrlevtxm1 | Structured version Visualization version GIF version |
Description: The number of neighbors of a vertex is at most the number of vertices of the graph minus 1 in a finite simple graph. (Contributed by AV, 16-Dec-2020.) (Proof shortened by AV, 13-Feb-2022.) |
Ref | Expression |
---|---|
hashnbusgrnn0.v | ⢠ð = (Vtxâðº) |
Ref | Expression |
---|---|
nbfusgrlevtxm1 | ⢠((ðº â FinUSGraph ⧠ð â ð) â (â¯â(ðº NeighbVtx ð)) †((â¯âð) â 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashnbusgrnn0.v | . . . . 5 ⢠ð = (Vtxâðº) | |
2 | 1 | fvexi 6861 | . . . 4 ⢠ð â V |
3 | 2 | difexi 5290 | . . 3 ⢠(ð â {ð}) â V |
4 | 1 | nbgrssovtx 28351 | . . . 4 ⢠(ðº NeighbVtx ð) â (ð â {ð}) |
5 | 4 | a1i 11 | . . 3 ⢠((ðº â FinUSGraph ⧠ð â ð) â (ðº NeighbVtx ð) â (ð â {ð})) |
6 | hashss 14316 | . . 3 ⢠(((ð â {ð}) â V ⧠(ðº NeighbVtx ð) â (ð â {ð})) â (â¯â(ðº NeighbVtx ð)) †(â¯â(ð â {ð}))) | |
7 | 3, 5, 6 | sylancr 588 | . 2 ⢠((ðº â FinUSGraph ⧠ð â ð) â (â¯â(ðº NeighbVtx ð)) †(â¯â(ð â {ð}))) |
8 | 1 | fusgrvtxfi 28309 | . . 3 ⢠(ðº â FinUSGraph â ð â Fin) |
9 | hashdifsn 14321 | . . . 4 ⢠((ð â Fin ⧠ð â ð) â (â¯â(ð â {ð})) = ((â¯âð) â 1)) | |
10 | 9 | eqcomd 2743 | . . 3 ⢠((ð â Fin ⧠ð â ð) â ((â¯âð) â 1) = (â¯â(ð â {ð}))) |
11 | 8, 10 | sylan 581 | . 2 ⢠((ðº â FinUSGraph ⧠ð â ð) â ((â¯âð) â 1) = (â¯â(ð â {ð}))) |
12 | 7, 11 | breqtrrd 5138 | 1 ⢠((ðº â FinUSGraph ⧠ð â ð) â (â¯â(ðº NeighbVtx ð)) †((â¯âð) â 1)) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 ⧠wa 397 = wceq 1542 â wcel 2107 Vcvv 3448 â cdif 3912 â wss 3915 {csn 4591 class class class wbr 5110 âcfv 6501 (class class class)co 7362 Fincfn 8890 1c1 11059 †cle 11197 â cmin 11392 â¯chash 14237 Vtxcvtx 27989 FinUSGraphcfusgr 28306 NeighbVtx cnbgr 28322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-xnn0 12493 df-z 12507 df-uz 12771 df-fz 13432 df-hash 14238 df-fusgr 28307 df-nbgr 28323 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |