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Theorem usgranloopv 23444
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 4098 . . . . . . . 8  |-  ( M  e.  W  ->  { M ,  N }  =/=  (/) )
21adantl 466 . . . . . . 7  |-  ( ( ( E `  X
)  =  { M ,  N }  /\  M  e.  W )  ->  { M ,  N }  =/=  (/) )
3 neeq1 2730 . . . . . . . 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 5826 . . . . . 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 23442 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( ( E `  X )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V
) ) )
9 usgraedg2 23441 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( # `  ( E `
 X ) )  =  2 )
10 fveq2 5794 . . . . . . . . . . . . . 14  |-  ( ( E `  X )  =  { M ,  N }  ->  ( # `  ( E `  X
) )  =  (
# `  { M ,  N } ) )
1110eqeq1d 2454 . . . . . . . . . . . . 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 12268 . . . . . . . . . . . . 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 1370    e. wcel 1758    =/= wne 2645   (/)c0 3740   {csn 3980   {cpr 3982   class class class wbr 4395   dom cdm 4943    |` cres 4945   Fun wfun 5515   ` cfv 5521   2c2 10477   #chash 12215   USGrph cusg 23411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-n0 10686  df-z 10753  df-uz 10968  df-fz 11550  df-hash 12216  df-usgra 23413
This theorem is referenced by:  usgranloop  23445  usgra2pthspth  30438  usgra2pthlem1  30443
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