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Theorem nb3gra2nb 23542
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.)
Assertion
Ref Expression
nb3gra2nb  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B )  =  { A ,  C }
)  <->  ( ( <. V ,  E >. Neighbors  A
)  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B
)  =  { A ,  C }  /\  ( <. V ,  E >. Neighbors  C
)  =  { A ,  B } ) ) )

Proof of Theorem nb3gra2nb
StepHypRef Expression
1 prcom 4064 . . . . . . . . 9  |-  { A ,  C }  =  { C ,  A }
21eleq1i 2531 . . . . . . . 8  |-  ( { A ,  C }  e.  ran  E  <->  { C ,  A }  e.  ran  E )
32biimpi 194 . . . . . . 7  |-  ( { A ,  C }  e.  ran  E  ->  { C ,  A }  e.  ran  E )
43adantl 466 . . . . . 6  |-  ( ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E )  ->  { C ,  A }  e.  ran  E )
5 prcom 4064 . . . . . . . . 9  |-  { B ,  C }  =  { C ,  B }
65eleq1i 2531 . . . . . . . 8  |-  ( { B ,  C }  e.  ran  E  <->  { C ,  B }  e.  ran  E )
76biimpi 194 . . . . . . 7  |-  ( { B ,  C }  e.  ran  E  ->  { C ,  B }  e.  ran  E )
87adantl 466 . . . . . 6  |-  ( ( { B ,  A }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  { C ,  B }  e.  ran  E )
94, 8anim12i 566 . . . . 5  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E )  /\  ( { B ,  A }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
109a1i 11 . . . 4  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E )  /\  ( { B ,  A }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
11 nb3graprlem1 23538 . . . . 5  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( <. V ,  E >. Neighbors  A )  =  { B ,  C }  <->  ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )
12 3ancoma 972 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  <->  ( B  e.  Y  /\  A  e.  X  /\  C  e.  Z )
)
1312biimpi 194 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( B  e.  Y  /\  A  e.  X  /\  C  e.  Z
) )
14 tpcoma 4082 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  A ,  C }
1514eqeq2i 2472 . . . . . . . 8  |-  ( V  =  { A ,  B ,  C }  <->  V  =  { B ,  A ,  C }
)
1615biimpi 194 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { B ,  A ,  C }
)
1716anim1i 568 . . . . . 6  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { B ,  A ,  C }  /\  V USGrph  E
) )
18 nb3graprlem1 23538 . . . . . 6  |-  ( ( ( B  e.  Y  /\  A  e.  X  /\  C  e.  Z
)  /\  ( V  =  { B ,  A ,  C }  /\  V USGrph  E ) )  ->  (
( <. V ,  E >. Neighbors  B )  =  { A ,  C }  <->  ( { B ,  A }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
1913, 17, 18syl2an 477 . . . . 5  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( <. V ,  E >. Neighbors  B )  =  { A ,  C }  <->  ( { B ,  A }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
2011, 19anbi12d 710 . . . 4  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B )  =  { A ,  C }
)  <->  ( ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E )  /\  ( { B ,  A }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
21 3anrot 970 . . . . . 6  |-  ( ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y )  <->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )
)
2221biimpri 206 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y
) )
23 tprot 4081 . . . . . . . . 9  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
2423eqcomi 2467 . . . . . . . 8  |-  { A ,  B ,  C }  =  { C ,  A ,  B }
2524eqeq2i 2472 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  <->  V  =  { C ,  A ,  B }
)
2625anbi1i 695 . . . . . 6  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  <->  ( V  =  { C ,  A ,  B }  /\  V USGrph  E ) )
2726biimpi 194 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { C ,  A ,  B }  /\  V USGrph  E
) )
28 nb3graprlem1 23538 . . . . 5  |-  ( ( ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y
)  /\  ( V  =  { C ,  A ,  B }  /\  V USGrph  E ) )  ->  (
( <. V ,  E >. Neighbors  C )  =  { A ,  B }  <->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
2922, 27, 28syl2an 477 . . . 4  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( <. V ,  E >. Neighbors  C )  =  { A ,  B }  <->  ( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
3010, 20, 293imtr4d 268 . . 3  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B )  =  { A ,  C }
)  ->  ( <. V ,  E >. Neighbors  C )  =  { A ,  B } ) )
3130pm4.71d 634 . 2  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B )  =  { A ,  C }
)  <->  ( ( (
<. V ,  E >. Neighbors  A
)  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B
)  =  { A ,  C } )  /\  ( <. V ,  E >. Neighbors  C )  =  { A ,  B }
) ) )
32 df-3an 967 . 2  |-  ( ( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B )  =  { A ,  C }  /\  ( <. V ,  E >. Neighbors  C )  =  { A ,  B }
)  <->  ( ( (
<. V ,  E >. Neighbors  A
)  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B
)  =  { A ,  C } )  /\  ( <. V ,  E >. Neighbors  C )  =  { A ,  B }
) )
3331, 32syl6bbr 263 1  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B )  =  { A ,  C }
)  <->  ( ( <. V ,  E >. Neighbors  A
)  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B
)  =  { A ,  C }  /\  ( <. V ,  E >. Neighbors  C
)  =  { A ,  B } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {cpr 3990   {ctp 3992   <.cop 3994   class class class wbr 4403   ran crn 4952  (class class class)co 6203   USGrph cusg 23443   Neighbors cnbgra 23508
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-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
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 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8224  df-cda 8452  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-hash 12225  df-usgra 23445  df-nbgra 23511
This theorem is referenced by: (None)
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