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Theorem usgranloop 21352
Description: In an undirected simple graph without loops, there is no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Proof shortened by Alexander van der Vekens, 20-Mar-2018.)
Assertion
Ref Expression
usgranloop  |-  ( V USGrph  E  ->  ( E. x  e.  dom  E ( E `
 x )  =  { M ,  N }  ->  M  =/=  N
) )
Distinct variable groups:    x, E    x, V    x, M    x, N

Proof of Theorem usgranloop
StepHypRef Expression
1 usgraedgprv 21349 . . . . 5  |-  ( ( V USGrph  E  /\  x  e.  dom  E )  -> 
( ( E `  x )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V
) ) )
21imp 419 . . . 4  |-  ( ( ( V USGrph  E  /\  x  e.  dom  E )  /\  ( E `  x )  =  { M ,  N }
)  ->  ( M  e.  V  /\  N  e.  V ) )
3 usgranloopv 21351 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  M  e.  V )  ->  (
( E `  x
)  =  { M ,  N }  ->  M  =/=  N ) )
43ex 424 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( M  e.  V  ->  ( ( E `  x )  =  { M ,  N }  ->  M  =/=  N
) ) )
54com23 74 . . . . . . . 8  |-  ( V USGrph  E  ->  ( ( E `
 x )  =  { M ,  N }  ->  ( M  e.  V  ->  M  =/=  N ) ) )
65adantr 452 . . . . . . 7  |-  ( ( V USGrph  E  /\  x  e.  dom  E )  -> 
( ( E `  x )  =  { M ,  N }  ->  ( M  e.  V  ->  M  =/=  N ) ) )
76imp 419 . . . . . 6  |-  ( ( ( V USGrph  E  /\  x  e.  dom  E )  /\  ( E `  x )  =  { M ,  N }
)  ->  ( M  e.  V  ->  M  =/= 
N ) )
87com12 29 . . . . 5  |-  ( M  e.  V  ->  (
( ( V USGrph  E  /\  x  e.  dom  E )  /\  ( E `
 x )  =  { M ,  N } )  ->  M  =/=  N ) )
98adantr 452 . . . 4  |-  ( ( M  e.  V  /\  N  e.  V )  ->  ( ( ( V USGrph  E  /\  x  e.  dom  E )  /\  ( E `
 x )  =  { M ,  N } )  ->  M  =/=  N ) )
102, 9mpcom 34 . . 3  |-  ( ( ( V USGrph  E  /\  x  e.  dom  E )  /\  ( E `  x )  =  { M ,  N }
)  ->  M  =/=  N )
1110ex 424 . 2  |-  ( ( V USGrph  E  /\  x  e.  dom  E )  -> 
( ( E `  x )  =  { M ,  N }  ->  M  =/=  N ) )
1211rexlimdva 2790 1  |-  ( V USGrph  E  ->  ( E. x  e.  dom  E ( E `
 x )  =  { M ,  N }  ->  M  =/=  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   {cpr 3775   class class class wbr 4172   dom cdm 4837   ` cfv 5413   USGrph cusg 21318
This theorem is referenced by:  usgranloop0  21353  usgraedgrn  21354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-hash 11574  df-usgra 21320
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