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Theorem isusgra0 23275
Description: The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
Assertion
Ref Expression
isusgra0  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } ) )
Distinct variable groups:    x, E    x, V    x, W    x, X

Proof of Theorem isusgra0
StepHypRef Expression
1 isusgra 23272 . 2  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
2 vex 2975 . . . . . . . . . . . . . 14  |-  x  e. 
_V
3 hasheq0 12131 . . . . . . . . . . . . . 14  |-  ( x  e.  _V  ->  (
( # `  x )  =  0  <->  x  =  (/) ) )
42, 3ax-mp 5 . . . . . . . . . . . . 13  |-  ( (
# `  x )  =  0  <->  x  =  (/) )
5 2ne0 10414 . . . . . . . . . . . . . . . . 17  |-  2  =/=  0
65a1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
# `  x )  =  0  ->  2  =/=  0 )
7 id 22 . . . . . . . . . . . . . . . 16  |-  ( (
# `  x )  =  0  ->  ( # `
 x )  =  0 )
86, 7neeqtrrd 2632 . . . . . . . . . . . . . . 15  |-  ( (
# `  x )  =  0  ->  2  =/=  ( # `  x
) )
98necomd 2695 . . . . . . . . . . . . . 14  |-  ( (
# `  x )  =  0  ->  ( # `
 x )  =/=  2 )
109neneqd 2624 . . . . . . . . . . . . 13  |-  ( (
# `  x )  =  0  ->  -.  ( # `  x )  =  2 )
114, 10sylbir 213 . . . . . . . . . . . 12  |-  ( x  =  (/)  ->  -.  ( # `
 x )  =  2 )
1211con2i 120 . . . . . . . . . . 11  |-  ( (
# `  x )  =  2  ->  -.  x  =  (/) )
1312adantl 466 . . . . . . . . . 10  |-  ( ( ( V  e.  W  /\  E  e.  X
)  /\  ( # `  x
)  =  2 )  ->  -.  x  =  (/) )
14 elsn 3891 . . . . . . . . . 10  |-  ( x  e.  { (/) }  <->  x  =  (/) )
1513, 14sylnibr 305 . . . . . . . . 9  |-  ( ( ( V  e.  W  /\  E  e.  X
)  /\  ( # `  x
)  =  2 )  ->  -.  x  e.  {
(/) } )
1615biantrud 507 . . . . . . . 8  |-  ( ( ( V  e.  W  /\  E  e.  X
)  /\  ( # `  x
)  =  2 )  ->  ( x  e. 
~P V  <->  ( x  e.  ~P V  /\  -.  x  e.  { (/) } ) ) )
17 eldif 3338 . . . . . . . 8  |-  ( x  e.  ( ~P V  \  { (/) } )  <->  ( x  e.  ~P V  /\  -.  x  e.  { (/) } ) )
1816, 17syl6rbbr 264 . . . . . . 7  |-  ( ( ( V  e.  W  /\  E  e.  X
)  /\  ( # `  x
)  =  2 )  ->  ( x  e.  ( ~P V  \  { (/) } )  <->  x  e.  ~P V ) )
1918ex 434 . . . . . 6  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( ( # `  x
)  =  2  -> 
( x  e.  ( ~P V  \  { (/)
} )  <->  x  e.  ~P V ) ) )
2019pm5.32rd 640 . . . . 5  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( ( x  e.  ( ~P V  \  { (/) } )  /\  ( # `  x )  =  2 )  <->  ( x  e.  ~P V  /\  ( # `
 x )  =  2 ) ) )
2120abbidv 2557 . . . 4  |-  ( ( V  e.  W  /\  E  e.  X )  ->  { x  |  ( x  e.  ( ~P V  \  { (/) } )  /\  ( # `  x )  =  2 ) }  =  {
x  |  ( x  e.  ~P V  /\  ( # `  x )  =  2 ) } )
22 df-rab 2724 . . . 4  |-  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 }  =  { x  |  ( x  e.  ( ~P V  \  { (/)
} )  /\  ( # `
 x )  =  2 ) }
23 df-rab 2724 . . . 4  |-  { x  e.  ~P V  |  (
# `  x )  =  2 }  =  { x  |  (
x  e.  ~P V  /\  ( # `  x
)  =  2 ) }
2421, 22, 233eqtr4g 2500 . . 3  |-  ( ( V  e.  W  /\  E  e.  X )  ->  { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 }  =  { x  e.  ~P V  |  ( # `  x
)  =  2 } )
25 f1eq3 5603 . . 3  |-  ( { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }  =  { x  e.  ~P V  |  (
# `  x )  =  2 }  ->  ( E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }  <->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } ) )
2624, 25syl 16 . 2  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( E : dom  E
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 }  <->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } ) )
271, 26bitrd 253 1  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429    =/= wne 2606   {crab 2719   _Vcvv 2972    \ cdif 3325   (/)c0 3637   ~Pcpw 3860   {csn 3877   class class class wbr 4292   dom cdm 4840   -1-1->wf1 5415   ` cfv 5418   0cc0 9282   2c2 10371   #chash 12103   USGrph cusg 23264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-hash 12104  df-usgra 23266
This theorem is referenced by:  usgraf0  23276  ausisusgra  23279  usgra1  23292  usgraexmpl  23319  usgrares1  23323  cusgraexilem2  23375
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