MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgranloopv Structured version   Unicode version

Theorem usgranloopv 24054
Description: In an undirected simple graph without loops, there is no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
Assertion
Ref Expression
usgranloopv  |-  ( ( V USGrph  E  /\  M  e.  W )  ->  (
( E `  X
)  =  { M ,  N }  ->  M  =/=  N ) )

Proof of Theorem usgranloopv
StepHypRef Expression
1 prnzg 4147 . . . . . . . 8  |-  ( M  e.  W  ->  { M ,  N }  =/=  (/) )
21adantl 466 . . . . . . 7  |-  ( ( ( E `  X
)  =  { M ,  N }  /\  M  e.  W )  ->  { M ,  N }  =/=  (/) )
3 neeq1 2748 . . . . . . . 8  |-  ( ( E `  X )  =  { M ,  N }  ->  ( ( E `  X )  =/=  (/)  <->  { M ,  N }  =/=  (/) ) )
43adantr 465 . . . . . . 7  |-  ( ( ( E `  X
)  =  { M ,  N }  /\  M  e.  W )  ->  (
( E `  X
)  =/=  (/)  <->  { M ,  N }  =/=  (/) ) )
52, 4mpbird 232 . . . . . 6  |-  ( ( ( E `  X
)  =  { M ,  N }  /\  M  e.  W )  ->  ( E `  X )  =/=  (/) )
6 fvfundmfvn0 5896 . . . . . 6  |-  ( ( E `  X )  =/=  (/)  ->  ( X  e.  dom  E  /\  Fun  ( E  |`  { X } ) ) )
75, 6syl 16 . . . . 5  |-  ( ( ( E `  X
)  =  { M ,  N }  /\  M  e.  W )  ->  ( X  e.  dom  E  /\  Fun  ( E  |`  { X } ) ) )
8 usgraedgprv 24052 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( ( E `  X )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V
) ) )
9 usgraedg2 24051 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( # `  ( E `
 X ) )  =  2 )
10 fveq2 5864 . . . . . . . . . . . . . 14  |-  ( ( E `  X )  =  { M ,  N }  ->  ( # `  ( E `  X
) )  =  (
# `  { M ,  N } ) )
1110eqeq1d 2469 . . . . . . . . . . . . 13  |-  ( ( E `  X )  =  { M ,  N }  ->  ( (
# `  ( E `  X ) )  =  2  <->  ( # `  { M ,  N }
)  =  2 ) )
1211biimpd 207 . . . . . . . . . . . 12  |-  ( ( E `  X )  =  { M ,  N }  ->  ( (
# `  ( E `  X ) )  =  2  ->  ( # `  { M ,  N }
)  =  2 ) )
13 hashprg 12424 . . . . . . . . . . . . 13  |-  ( ( M  e.  V  /\  N  e.  V )  ->  ( M  =/=  N  <->  (
# `  { M ,  N } )  =  2 ) )
1413biimprcd 225 . . . . . . . . . . . 12  |-  ( (
# `  { M ,  N } )  =  2  ->  ( ( M  e.  V  /\  N  e.  V )  ->  M  =/=  N ) )
1512, 14syl6com 35 . . . . . . . . . . 11  |-  ( (
# `  ( E `  X ) )  =  2  ->  ( ( E `  X )  =  { M ,  N }  ->  ( ( M  e.  V  /\  N  e.  V )  ->  M  =/=  N ) ) )
169, 15syl 16 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( ( E `  X )  =  { M ,  N }  ->  ( ( M  e.  V  /\  N  e.  V )  ->  M  =/=  N ) ) )
178, 16mpdd 40 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( ( E `  X )  =  { M ,  N }  ->  M  =/=  N ) )
1817adantrd 468 . . . . . . . 8  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( ( ( E `
 X )  =  { M ,  N }  /\  M  e.  W
)  ->  M  =/=  N ) )
1918expcom 435 . . . . . . 7  |-  ( X  e.  dom  E  -> 
( V USGrph  E  ->  ( ( ( E `  X )  =  { M ,  N }  /\  M  e.  W
)  ->  M  =/=  N ) ) )
2019com23 78 . . . . . 6  |-  ( X  e.  dom  E  -> 
( ( ( E `
 X )  =  { M ,  N }  /\  M  e.  W
)  ->  ( V USGrph  E  ->  M  =/=  N
) ) )
2120adantr 465 . . . . 5  |-  ( ( X  e.  dom  E  /\  Fun  ( E  |`  { X } ) )  ->  ( ( ( E `  X )  =  { M ,  N }  /\  M  e.  W )  ->  ( V USGrph  E  ->  M  =/=  N ) ) )
227, 21mpcom 36 . . . 4  |-  ( ( ( E `  X
)  =  { M ,  N }  /\  M  e.  W )  ->  ( V USGrph  E  ->  M  =/=  N ) )
2322ex 434 . . 3  |-  ( ( E `  X )  =  { M ,  N }  ->  ( M  e.  W  ->  ( V USGrph  E  ->  M  =/=  N ) ) )
2423com13 80 . 2  |-  ( V USGrph  E  ->  ( M  e.  W  ->  ( ( E `  X )  =  { M ,  N }  ->  M  =/=  N
) ) )
2524imp 429 1  |-  ( ( V USGrph  E  /\  M  e.  W )  ->  (
( E `  X
)  =  { M ,  N }  ->  M  =/=  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   (/)c0 3785   {csn 4027   {cpr 4029   class class class wbr 4447   dom cdm 4999    |` cres 5001   Fun wfun 5580   ` cfv 5586   2c2 10581   #chash 12369   USGrph cusg 24006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-hash 12370  df-usgra 24009
This theorem is referenced by:  usgranloop  24055  usgra2pthspth  31820  usgra2pthlem1  31822
  Copyright terms: Public domain W3C validator