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Theorem usgpredgv 32771
Description: An edge of a graph always connects two vertices, analogous to usgraedgrnv 24579. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 9-Jan-2020.)
Hypotheses
Ref Expression
usgedgppr.e  |-  E  =  ( Edges  `  G )
usgpredgv.v  |-  V  =  ( Vtx  `  G )
Assertion
Ref Expression
usgpredgv  |-  ( ( G  e. USGrph  /\  { M ,  N }  e.  E
)  ->  ( M  e.  V  /\  N  e.  V ) )

Proof of Theorem usgpredgv
StepHypRef Expression
1 usgedgppr.e . . . 4  |-  E  =  ( Edges  `  G )
21eleq2i 2532 . . 3  |-  ( { M ,  N }  e.  E  <->  { M ,  N }  e.  ( Edges  `  G
) )
3 edg 24555 . . 3  |-  ( ( G  e. USGrph  /\  { M ,  N }  e.  ( Edges  `  G ) )  -> 
( { M ,  N }  e.  ~P ( 1st `  G )  /\  ( # `  { M ,  N }
)  =  2 ) )
42, 3sylan2b 473 . 2  |-  ( ( G  e. USGrph  /\  { M ,  N }  e.  E
)  ->  ( { M ,  N }  e.  ~P ( 1st `  G
)  /\  ( # `  { M ,  N }
)  =  2 ) )
5 hashprb 12446 . . . . . . . . 9  |-  ( ( M  e.  _V  /\  N  e.  _V  /\  M  =/=  N )  <->  ( # `  { M ,  N }
)  =  2 )
6 usgpredgv.v . . . . . . . . . . . . . . 15  |-  V  =  ( Vtx  `  G )
7 vtxval 32755 . . . . . . . . . . . . . . 15  |-  ( G  e. USGrph  ->  ( Vtx  `  G
)  =  ( 1st `  G ) )
86, 7syl5eq 2507 . . . . . . . . . . . . . 14  |-  ( G  e. USGrph  ->  V  =  ( 1st `  G ) )
9 eleq2 2527 . . . . . . . . . . . . . . 15  |-  ( V  =  ( 1st `  G
)  ->  ( M  e.  V  <->  M  e.  ( 1st `  G ) ) )
10 eleq2 2527 . . . . . . . . . . . . . . 15  |-  ( V  =  ( 1st `  G
)  ->  ( N  e.  V  <->  N  e.  ( 1st `  G ) ) )
119, 10anbi12d 708 . . . . . . . . . . . . . 14  |-  ( V  =  ( 1st `  G
)  ->  ( ( M  e.  V  /\  N  e.  V )  <->  ( M  e.  ( 1st `  G )  /\  N  e.  ( 1st `  G
) ) ) )
128, 11syl 16 . . . . . . . . . . . . 13  |-  ( G  e. USGrph  ->  ( ( M  e.  V  /\  N  e.  V )  <->  ( M  e.  ( 1st `  G
)  /\  N  e.  ( 1st `  G ) ) ) )
13 prelpw 32673 . . . . . . . . . . . . 13  |-  ( ( M  e.  _V  /\  N  e.  _V )  ->  ( ( M  e.  ( 1st `  G
)  /\  N  e.  ( 1st `  G ) )  <->  { M ,  N }  e.  ~P ( 1st `  G ) ) )
1412, 13sylan9bbr 698 . . . . . . . . . . . 12  |-  ( ( ( M  e.  _V  /\  N  e.  _V )  /\  G  e. USGrph  )  -> 
( ( M  e.  V  /\  N  e.  V )  <->  { M ,  N }  e.  ~P ( 1st `  G ) ) )
1514bicomd 201 . . . . . . . . . . 11  |-  ( ( ( M  e.  _V  /\  N  e.  _V )  /\  G  e. USGrph  )  -> 
( { M ,  N }  e.  ~P ( 1st `  G )  <-> 
( M  e.  V  /\  N  e.  V
) ) )
1615ex 432 . . . . . . . . . 10  |-  ( ( M  e.  _V  /\  N  e.  _V )  ->  ( G  e. USGrph  ->  ( { M ,  N }  e.  ~P ( 1st `  G )  <->  ( M  e.  V  /\  N  e.  V ) ) ) )
17163adant3 1014 . . . . . . . . 9  |-  ( ( M  e.  _V  /\  N  e.  _V  /\  M  =/=  N )  ->  ( G  e. USGrph  ->  ( { M ,  N }  e.  ~P ( 1st `  G
)  <->  ( M  e.  V  /\  N  e.  V ) ) ) )
185, 17sylbir 213 . . . . . . . 8  |-  ( (
# `  { M ,  N } )  =  2  ->  ( G  e. USGrph  ->  ( { M ,  N }  e.  ~P ( 1st `  G )  <-> 
( M  e.  V  /\  N  e.  V
) ) ) )
1918impcom 428 . . . . . . 7  |-  ( ( G  e. USGrph  /\  ( # `
 { M ,  N } )  =  2 )  ->  ( { M ,  N }  e.  ~P ( 1st `  G
)  <->  ( M  e.  V  /\  N  e.  V ) ) )
2019biimpd 207 . . . . . 6  |-  ( ( G  e. USGrph  /\  ( # `
 { M ,  N } )  =  2 )  ->  ( { M ,  N }  e.  ~P ( 1st `  G
)  ->  ( M  e.  V  /\  N  e.  V ) ) )
2120ex 432 . . . . 5  |-  ( G  e. USGrph  ->  ( ( # `  { M ,  N } )  =  2  ->  ( { M ,  N }  e.  ~P ( 1st `  G )  ->  ( M  e.  V  /\  N  e.  V ) ) ) )
2221com23 78 . . . 4  |-  ( G  e. USGrph  ->  ( { M ,  N }  e.  ~P ( 1st `  G )  ->  ( ( # `  { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) ) )
2322impd 429 . . 3  |-  ( G  e. USGrph  ->  ( ( { M ,  N }  e.  ~P ( 1st `  G
)  /\  ( # `  { M ,  N }
)  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) ) )
2423adantr 463 . 2  |-  ( ( G  e. USGrph  /\  { M ,  N }  e.  E
)  ->  ( ( { M ,  N }  e.  ~P ( 1st `  G
)  /\  ( # `  { M ,  N }
)  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) ) )
254, 24mpd 15 1  |-  ( ( G  e. USGrph  /\  { M ,  N }  e.  E
)  ->  ( M  e.  V  /\  N  e.  V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106   ~Pcpw 3999   {cpr 4018   ` cfv 5570   1stc1st 6771   2c2 10581   #chash 12387   USGrph cusg 24532   Edges cedg 24533   Vtx cvtx 32753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-hash 12388  df-usgra 24535  df-edg 24538  df-vtx 32754
This theorem is referenced by:  usgedgimp  32775  usgpredgdv  32781  usgvad2edg  32783
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