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Theorem usgranloopv 24782
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 4091 . . . . . . . 8  |-  ( M  e.  W  ->  { M ,  N }  =/=  (/) )
21adantl 464 . . . . . . 7  |-  ( ( ( E `  X
)  =  { M ,  N }  /\  M  e.  W )  ->  { M ,  N }  =/=  (/) )
3 neeq1 2684 . . . . . . . 8  |-  ( ( E `  X )  =  { M ,  N }  ->  ( ( E `  X )  =/=  (/)  <->  { M ,  N }  =/=  (/) ) )
43adantr 463 . . . . . . 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 5880 . . . . . 6  |-  ( ( E `  X )  =/=  (/)  ->  ( X  e.  dom  E  /\  Fun  ( E  |`  { X } ) ) )
75, 6syl 17 . . . . 5  |-  ( ( ( E `  X
)  =  { M ,  N }  /\  M  e.  W )  ->  ( X  e.  dom  E  /\  Fun  ( E  |`  { X } ) ) )
8 usgraedgprv 24780 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( ( E `  X )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V
) ) )
9 usgraedg2 24779 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( # `  ( E `
 X ) )  =  2 )
10 fveq2 5848 . . . . . . . . . . . . . 14  |-  ( ( E `  X )  =  { M ,  N }  ->  ( # `  ( E `  X
) )  =  (
# `  { M ,  N } ) )
1110eqeq1d 2404 . . . . . . . . . . . . 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 12507 . . . . . . . . . . . . 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 33 . . . . . . . . . . 11  |-  ( (
# `  ( E `  X ) )  =  2  ->  ( ( E `  X )  =  { M ,  N }  ->  ( ( M  e.  V  /\  N  e.  V )  ->  M  =/=  N ) ) )
169, 15syl 17 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( ( E `  X )  =  { M ,  N }  ->  ( ( M  e.  V  /\  N  e.  V )  ->  M  =/=  N ) ) )
178, 16mpdd 38 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( ( E `  X )  =  { M ,  N }  ->  M  =/=  N ) )
1817adantrd 466 . . . . . . . 8  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( ( ( E `
 X )  =  { M ,  N }  /\  M  e.  W
)  ->  M  =/=  N ) )
1918expcom 433 . . . . . . 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 463 . . . . 5  |-  ( ( X  e.  dom  E  /\  Fun  ( E  |`  { X } ) )  ->  ( ( ( E `  X )  =  { M ,  N }  /\  M  e.  W )  ->  ( V USGrph  E  ->  M  =/=  N ) ) )
227, 21mpcom 34 . . . 4  |-  ( ( ( E `  X
)  =  { M ,  N }  /\  M  e.  W )  ->  ( V USGrph  E  ->  M  =/=  N ) )
2322ex 432 . . 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 427 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 367    = wceq 1405    e. wcel 1842    =/= wne 2598   (/)c0 3737   {csn 3971   {cpr 3973   class class class wbr 4394   dom cdm 4822    |` cres 4824   Fun wfun 5562   ` cfv 5568   2c2 10625   #chash 12450   USGrph cusg 24734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-hash 12451  df-usgra 24737
This theorem is referenced by:  usgranloop  24783  usgra2pthspth  37961  usgra2pthlem1  37963
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