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Theorem nb3gr2nb 39622
Description: If the neighbors of two vertices in a graph with three elements are an unordered pair of the other vertices, the neighbors of all three vertices are an unordered pair of the other vertices. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.)
Assertion
Ref Expression
nb3gr2nb  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( (Vtx `  G )  =  { A ,  B ,  C }  /\  G  e. USGraph  ) )  ->  (
( ( G NeighbVtx  A )  =  { B ,  C }  /\  ( G NeighbVtx  B )  =  { A ,  C }
)  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  /\  ( G NeighbVtx  B )  =  { A ,  C }  /\  ( G NeighbVtx  C )  =  { A ,  B } ) ) )

Proof of Theorem nb3gr2nb
StepHypRef Expression
1 prcom 4041 . . . . . . . . 9  |-  { A ,  C }  =  { C ,  A }
21eleq1i 2540 . . . . . . . 8  |-  ( { A ,  C }  e.  (Edg `  G )  <->  { C ,  A }  e.  (Edg `  G )
)
32biimpi 199 . . . . . . 7  |-  ( { A ,  C }  e.  (Edg `  G )  ->  { C ,  A }  e.  (Edg `  G
) )
43adantl 473 . . . . . 6  |-  ( ( { A ,  B }  e.  (Edg `  G
)  /\  { A ,  C }  e.  (Edg
`  G ) )  ->  { C ,  A }  e.  (Edg `  G ) )
5 prcom 4041 . . . . . . . . 9  |-  { B ,  C }  =  { C ,  B }
65eleq1i 2540 . . . . . . . 8  |-  ( { B ,  C }  e.  (Edg `  G )  <->  { C ,  B }  e.  (Edg `  G )
)
76biimpi 199 . . . . . . 7  |-  ( { B ,  C }  e.  (Edg `  G )  ->  { C ,  B }  e.  (Edg `  G
) )
87adantl 473 . . . . . 6  |-  ( ( { B ,  A }  e.  (Edg `  G
)  /\  { B ,  C }  e.  (Edg
`  G ) )  ->  { C ,  B }  e.  (Edg `  G ) )
94, 8anim12i 576 . . . . 5  |-  ( ( ( { A ,  B }  e.  (Edg `  G )  /\  { A ,  C }  e.  (Edg `  G )
)  /\  ( { B ,  A }  e.  (Edg `  G )  /\  { B ,  C }  e.  (Edg `  G
) ) )  -> 
( { C ,  A }  e.  (Edg `  G )  /\  { C ,  B }  e.  (Edg `  G )
) )
109a1i 11 . . . 4  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( (Vtx `  G )  =  { A ,  B ,  C }  /\  G  e. USGraph  ) )  ->  (
( ( { A ,  B }  e.  (Edg
`  G )  /\  { A ,  C }  e.  (Edg `  G )
)  /\  ( { B ,  A }  e.  (Edg `  G )  /\  { B ,  C }  e.  (Edg `  G
) ) )  -> 
( { C ,  A }  e.  (Edg `  G )  /\  { C ,  B }  e.  (Edg `  G )
) ) )
11 eqid 2471 . . . . . 6  |-  (Vtx `  G )  =  (Vtx
`  G )
12 eqid 2471 . . . . . 6  |-  (Edg `  G )  =  (Edg
`  G )
13 simprr 774 . . . . . 6  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( (Vtx `  G )  =  { A ,  B ,  C }  /\  G  e. USGraph  ) )  ->  G  e. USGraph  )
14 simprl 772 . . . . . 6  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( (Vtx `  G )  =  { A ,  B ,  C }  /\  G  e. USGraph  ) )  ->  (Vtx `  G )  =  { A ,  B ,  C } )
15 simpl 464 . . . . . 6  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( (Vtx `  G )  =  { A ,  B ,  C }  /\  G  e. USGraph  ) )  ->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )
)
1611, 12, 13, 14, 15nb3grprlem1 39618 . . . . 5  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( (Vtx `  G )  =  { A ,  B ,  C }  /\  G  e. USGraph  ) )  ->  (
( G NeighbVtx  A )  =  { B ,  C } 
<->  ( { A ,  B }  e.  (Edg `  G )  /\  { A ,  C }  e.  (Edg `  G )
) ) )
17 3ancoma 1014 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  <->  ( B  e.  Y  /\  A  e.  X  /\  C  e.  Z )
)
1817biimpi 199 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( B  e.  Y  /\  A  e.  X  /\  C  e.  Z
) )
19 tpcoma 4059 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  A ,  C }
2019eqeq2i 2483 . . . . . . . 8  |-  ( (Vtx
`  G )  =  { A ,  B ,  C }  <->  (Vtx `  G
)  =  { B ,  A ,  C }
)
2120biimpi 199 . . . . . . 7  |-  ( (Vtx
`  G )  =  { A ,  B ,  C }  ->  (Vtx `  G )  =  { B ,  A ,  C } )
2221anim1i 578 . . . . . 6  |-  ( ( (Vtx `  G )  =  { A ,  B ,  C }  /\  G  e. USGraph  )  ->  ( (Vtx `  G )  =  { B ,  A ,  C }  /\  G  e. USGraph  ) )
23 simprr 774 . . . . . . 7  |-  ( ( ( B  e.  Y  /\  A  e.  X  /\  C  e.  Z
)  /\  ( (Vtx `  G )  =  { B ,  A ,  C }  /\  G  e. USGraph  ) )  ->  G  e. USGraph  )
24 simprl 772 . . . . . . 7  |-  ( ( ( B  e.  Y  /\  A  e.  X  /\  C  e.  Z
)  /\  ( (Vtx `  G )  =  { B ,  A ,  C }  /\  G  e. USGraph  ) )  ->  (Vtx `  G )  =  { B ,  A ,  C } )
25 simpl 464 . . . . . . 7  |-  ( ( ( B  e.  Y  /\  A  e.  X  /\  C  e.  Z
)  /\  ( (Vtx `  G )  =  { B ,  A ,  C }  /\  G  e. USGraph  ) )  ->  ( B  e.  Y  /\  A  e.  X  /\  C  e.  Z )
)
2611, 12, 23, 24, 25nb3grprlem1 39618 . . . . . 6  |-  ( ( ( B  e.  Y  /\  A  e.  X  /\  C  e.  Z
)  /\  ( (Vtx `  G )  =  { B ,  A ,  C }  /\  G  e. USGraph  ) )  ->  (
( G NeighbVtx  B )  =  { A ,  C } 
<->  ( { B ,  A }  e.  (Edg `  G )  /\  { B ,  C }  e.  (Edg `  G )
) ) )
2718, 22, 26syl2an 485 . . . . 5  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( (Vtx `  G )  =  { A ,  B ,  C }  /\  G  e. USGraph  ) )  ->  (
( G NeighbVtx  B )  =  { A ,  C } 
<->  ( { B ,  A }  e.  (Edg `  G )  /\  { B ,  C }  e.  (Edg `  G )
) ) )
2816, 27anbi12d 725 . . . 4  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( (Vtx `  G )  =  { A ,  B ,  C }  /\  G  e. USGraph  ) )  ->  (
( ( G NeighbVtx  A )  =  { B ,  C }  /\  ( G NeighbVtx  B )  =  { A ,  C }
)  <->  ( ( { A ,  B }  e.  (Edg `  G )  /\  { A ,  C }  e.  (Edg `  G
) )  /\  ( { B ,  A }  e.  (Edg `  G )  /\  { B ,  C }  e.  (Edg `  G
) ) ) ) )
29 3anrot 1012 . . . . . 6  |-  ( ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y )  <->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )
)
3029biimpri 211 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y
) )
31 tprot 4058 . . . . . . . . 9  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
3231eqcomi 2480 . . . . . . . 8  |-  { A ,  B ,  C }  =  { C ,  A ,  B }
3332eqeq2i 2483 . . . . . . 7  |-  ( (Vtx
`  G )  =  { A ,  B ,  C }  <->  (Vtx `  G
)  =  { C ,  A ,  B }
)
3433anbi1i 709 . . . . . 6  |-  ( ( (Vtx `  G )  =  { A ,  B ,  C }  /\  G  e. USGraph  )  <->  ( (Vtx `  G )  =  { C ,  A ,  B }  /\  G  e. USGraph  ) )
3534biimpi 199 . . . . 5  |-  ( ( (Vtx `  G )  =  { A ,  B ,  C }  /\  G  e. USGraph  )  ->  ( (Vtx `  G )  =  { C ,  A ,  B }  /\  G  e. USGraph  ) )
36 simprr 774 . . . . . 6  |-  ( ( ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y
)  /\  ( (Vtx `  G )  =  { C ,  A ,  B }  /\  G  e. USGraph  ) )  ->  G  e. USGraph  )
37 simprl 772 . . . . . 6  |-  ( ( ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y
)  /\  ( (Vtx `  G )  =  { C ,  A ,  B }  /\  G  e. USGraph  ) )  ->  (Vtx `  G )  =  { C ,  A ,  B } )
38 simpl 464 . . . . . 6  |-  ( ( ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y
)  /\  ( (Vtx `  G )  =  { C ,  A ,  B }  /\  G  e. USGraph  ) )  ->  ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y )
)
3911, 12, 36, 37, 38nb3grprlem1 39618 . . . . 5  |-  ( ( ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y
)  /\  ( (Vtx `  G )  =  { C ,  A ,  B }  /\  G  e. USGraph  ) )  ->  (
( G NeighbVtx  C )  =  { A ,  B } 
<->  ( { C ,  A }  e.  (Edg `  G )  /\  { C ,  B }  e.  (Edg `  G )
) ) )
4030, 35, 39syl2an 485 . . . 4  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( (Vtx `  G )  =  { A ,  B ,  C }  /\  G  e. USGraph  ) )  ->  (
( G NeighbVtx  C )  =  { A ,  B } 
<->  ( { C ,  A }  e.  (Edg `  G )  /\  { C ,  B }  e.  (Edg `  G )
) ) )
4110, 28, 403imtr4d 276 . . 3  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( (Vtx `  G )  =  { A ,  B ,  C }  /\  G  e. USGraph  ) )  ->  (
( ( G NeighbVtx  A )  =  { B ,  C }  /\  ( G NeighbVtx  B )  =  { A ,  C }
)  ->  ( G NeighbVtx  C )  =  { A ,  B } ) )
4241pm4.71d 646 . 2  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( (Vtx `  G )  =  { A ,  B ,  C }  /\  G  e. USGraph  ) )  ->  (
( ( G NeighbVtx  A )  =  { B ,  C }  /\  ( G NeighbVtx  B )  =  { A ,  C }
)  <->  ( ( ( G NeighbVtx  A )  =  { B ,  C }  /\  ( G NeighbVtx  B )  =  { A ,  C } )  /\  ( G NeighbVtx  C )  =  { A ,  B }
) ) )
43 df-3an 1009 . 2  |-  ( ( ( G NeighbVtx  A )  =  { B ,  C }  /\  ( G NeighbVtx  B )  =  { A ,  C }  /\  ( G NeighbVtx  C )  =  { A ,  B }
)  <->  ( ( ( G NeighbVtx  A )  =  { B ,  C }  /\  ( G NeighbVtx  B )  =  { A ,  C } )  /\  ( G NeighbVtx  C )  =  { A ,  B }
) )
4442, 43syl6bbr 271 1  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( (Vtx `  G )  =  { A ,  B ,  C }  /\  G  e. USGraph  ) )  ->  (
( ( G NeighbVtx  A )  =  { B ,  C }  /\  ( G NeighbVtx  B )  =  { A ,  C }
)  <->  ( ( G NeighbVtx  A )  =  { B ,  C }  /\  ( G NeighbVtx  B )  =  { A ,  C }  /\  ( G NeighbVtx  C )  =  { A ,  B } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   {cpr 3961   {ctp 3963   ` cfv 5589  (class class class)co 6308  Vtxcvtx 39251  Edgcedga 39371   USGraph cusgr 39397   NeighbVtx cnbgr 39561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-upgr 39328  df-umgr 39329  df-edga 39372  df-usgr 39399  df-nbgr 39565
This theorem is referenced by: (None)
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