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Theorem usgraedgrnv 23447
Description: An edge of an undirected simple graph always connects two vertices. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
Assertion
Ref Expression
usgraedgrnv  |-  ( ( V USGrph  E  /\  { M ,  N }  e.  ran  E )  ->  ( M  e.  V  /\  N  e.  V ) )

Proof of Theorem usgraedgrnv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 usgraf0 23427 . . 3  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } )
2 f1f 5713 . . 3  |-  ( E : dom  E -1-1-> {
x  e.  ~P V  |  ( # `  x
)  =  2 }  ->  E : dom  E --> { x  e.  ~P V  |  ( # `  x
)  =  2 } )
3 df-f 5529 . . . 4  |-  ( E : dom  E --> { x  e.  ~P V  |  (
# `  x )  =  2 }  <->  ( E  Fn  dom  E  /\  ran  E 
C_  { x  e. 
~P V  |  (
# `  x )  =  2 } ) )
4 ssel2 3458 . . . . . . 7  |-  ( ( ran  E  C_  { x  e.  ~P V  |  (
# `  x )  =  2 }  /\  { M ,  N }  e.  ran  E )  ->  { M ,  N }  e.  { x  e.  ~P V  |  ( # `  x
)  =  2 } )
5 fveq2 5798 . . . . . . . . . 10  |-  ( x  =  { M ,  N }  ->  ( # `  x )  =  (
# `  { M ,  N } ) )
65eqeq1d 2456 . . . . . . . . 9  |-  ( x  =  { M ,  N }  ->  ( (
# `  x )  =  2  <->  ( # `  { M ,  N }
)  =  2 ) )
76elrab 3222 . . . . . . . 8  |-  ( { M ,  N }  e.  { x  e.  ~P V  |  ( # `  x
)  =  2 }  <-> 
( { M ,  N }  e.  ~P V  /\  ( # `  { M ,  N }
)  =  2 ) )
8 prex 4641 . . . . . . . . . . 11  |-  { M ,  N }  e.  _V
98elpw 3973 . . . . . . . . . 10  |-  ( { M ,  N }  e.  ~P V  <->  { M ,  N }  C_  V
)
10 ianor 488 . . . . . . . . . . . . . 14  |-  ( -.  ( M  e.  _V  /\  N  e.  _V )  <->  ( -.  M  e.  _V  \/  -.  N  e.  _V ) )
11 elprchashprn2 12273 . . . . . . . . . . . . . . 15  |-  ( -.  M  e.  _V  ->  -.  ( # `  { M ,  N }
)  =  2 )
12 elprchashprn2 12273 . . . . . . . . . . . . . . . 16  |-  ( -.  N  e.  _V  ->  -.  ( # `  { N ,  M }
)  =  2 )
13 prcom 4060 . . . . . . . . . . . . . . . . . . 19  |-  { N ,  M }  =  { M ,  N }
1413a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( -.  N  e.  _V  ->  { N ,  M }  =  { M ,  N } )
1514fveq2d 5802 . . . . . . . . . . . . . . . . 17  |-  ( -.  N  e.  _V  ->  (
# `  { N ,  M } )  =  ( # `  { M ,  N }
) )
1615eqeq1d 2456 . . . . . . . . . . . . . . . 16  |-  ( -.  N  e.  _V  ->  ( ( # `  { N ,  M }
)  =  2  <->  ( # `
 { M ,  N } )  =  2 ) )
1712, 16mtbid 300 . . . . . . . . . . . . . . 15  |-  ( -.  N  e.  _V  ->  -.  ( # `  { M ,  N }
)  =  2 )
1811, 17jaoi 379 . . . . . . . . . . . . . 14  |-  ( ( -.  M  e.  _V  \/  -.  N  e.  _V )  ->  -.  ( # `  { M ,  N }
)  =  2 )
1910, 18sylbi 195 . . . . . . . . . . . . 13  |-  ( -.  ( M  e.  _V  /\  N  e.  _V )  ->  -.  ( # `  { M ,  N }
)  =  2 )
2019con4i 130 . . . . . . . . . . . 12  |-  ( (
# `  { M ,  N } )  =  2  ->  ( M  e.  _V  /\  N  e. 
_V ) )
21 prssg 4135 . . . . . . . . . . . . 13  |-  ( ( M  e.  _V  /\  N  e.  _V )  ->  ( ( M  e.  V  /\  N  e.  V )  <->  { M ,  N }  C_  V
) )
2221biimprd 223 . . . . . . . . . . . 12  |-  ( ( M  e.  _V  /\  N  e.  _V )  ->  ( { M ,  N }  C_  V  -> 
( M  e.  V  /\  N  e.  V
) ) )
2320, 22syl 16 . . . . . . . . . . 11  |-  ( (
# `  { M ,  N } )  =  2  ->  ( { M ,  N }  C_  V  ->  ( M  e.  V  /\  N  e.  V ) ) )
2423com12 31 . . . . . . . . . 10  |-  ( { M ,  N }  C_  V  ->  ( ( # `
 { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
259, 24sylbi 195 . . . . . . . . 9  |-  ( { M ,  N }  e.  ~P V  ->  (
( # `  { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
2625imp 429 . . . . . . . 8  |-  ( ( { M ,  N }  e.  ~P V  /\  ( # `  { M ,  N }
)  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) )
277, 26sylbi 195 . . . . . . 7  |-  ( { M ,  N }  e.  { x  e.  ~P V  |  ( # `  x
)  =  2 }  ->  ( M  e.  V  /\  N  e.  V ) )
284, 27syl 16 . . . . . 6  |-  ( ( ran  E  C_  { x  e.  ~P V  |  (
# `  x )  =  2 }  /\  { M ,  N }  e.  ran  E )  -> 
( M  e.  V  /\  N  e.  V
) )
2928ex 434 . . . . 5  |-  ( ran 
E  C_  { x  e.  ~P V  |  (
# `  x )  =  2 }  ->  ( { M ,  N }  e.  ran  E  -> 
( M  e.  V  /\  N  e.  V
) ) )
3029adantl 466 . . . 4  |-  ( ( E  Fn  dom  E  /\  ran  E  C_  { x  e.  ~P V  |  (
# `  x )  =  2 } )  ->  ( { M ,  N }  e.  ran  E  ->  ( M  e.  V  /\  N  e.  V ) ) )
313, 30sylbi 195 . . 3  |-  ( E : dom  E --> { x  e.  ~P V  |  (
# `  x )  =  2 }  ->  ( { M ,  N }  e.  ran  E  -> 
( M  e.  V  /\  N  e.  V
) ) )
321, 2, 313syl 20 . 2  |-  ( V USGrph  E  ->  ( { M ,  N }  e.  ran  E  ->  ( M  e.  V  /\  N  e.  V ) ) )
3332imp 429 1  |-  ( ( V USGrph  E  /\  { M ,  N }  e.  ran  E )  ->  ( M  e.  V  /\  N  e.  V ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2802   _Vcvv 3076    C_ wss 3435   ~Pcpw 3967   {cpr 3986   class class class wbr 4399   dom cdm 4947   ran crn 4948    Fn wfn 5520   -->wf 5521   -1-1->wf1 5522   ` cfv 5525   2c2 10481   #chash 12219   USGrph cusg 23415
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 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-hash 12220  df-usgra 23417
This theorem is referenced by:  nbusgra  23490  nbgra0nb  23491  nbgraeledg  23492  nbgraisvtx  23493  constr3trllem2  23688  constr3trllem5  23691  usgra2adedgspthlem2  30451  usgra2adedgspth  30452  usgra2adedgwlk  30453  usgra2adedgwlkon  30454  frgranbnb  30759  frgraeu  30794  extwwlkfablem1  30814  numclwwlkun  30819
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