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Theorem nb3gra2nb 25159
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 4072 . . . . . . . . 9  |-  { A ,  C }  =  { C ,  A }
21eleq1i 2497 . . . . . . . 8  |-  ( { A ,  C }  e.  ran  E  <->  { C ,  A }  e.  ran  E )
32biimpi 197 . . . . . . 7  |-  ( { A ,  C }  e.  ran  E  ->  { C ,  A }  e.  ran  E )
43adantl 467 . . . . . 6  |-  ( ( { A ,  B }  e.  ran  E  /\  { A ,  C }  e.  ran  E )  ->  { C ,  A }  e.  ran  E )
5 prcom 4072 . . . . . . . . 9  |-  { B ,  C }  =  { C ,  B }
65eleq1i 2497 . . . . . . . 8  |-  ( { B ,  C }  e.  ran  E  <->  { C ,  B }  e.  ran  E )
76biimpi 197 . . . . . . 7  |-  ( { B ,  C }  e.  ran  E  ->  { C ,  B }  e.  ran  E )
87adantl 467 . . . . . 6  |-  ( ( { B ,  A }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  { C ,  B }  e.  ran  E )
94, 8anim12i 568 . . . . 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 25155 . . . . 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 989 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  <->  ( B  e.  Y  /\  A  e.  X  /\  C  e.  Z )
)
1312biimpi 197 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( B  e.  Y  /\  A  e.  X  /\  C  e.  Z
) )
14 tpcoma 4090 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  A ,  C }
1514eqeq2i 2438 . . . . . . . 8  |-  ( V  =  { A ,  B ,  C }  <->  V  =  { B ,  A ,  C }
)
1615biimpi 197 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { B ,  A ,  C }
)
1716anim1i 570 . . . . . 6  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { B ,  A ,  C }  /\  V USGrph  E
) )
18 nb3graprlem1 25155 . . . . . 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 479 . . . . 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 715 . . . 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 987 . . . . . 6  |-  ( ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y )  <->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )
)
2221biimpri 209 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y
) )
23 tprot 4089 . . . . . . . . 9  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
2423eqcomi 2433 . . . . . . . 8  |-  { A ,  B ,  C }  =  { C ,  A ,  B }
2524eqeq2i 2438 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  <->  V  =  { C ,  A ,  B }
)
2625anbi1i 699 . . . . . 6  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  <->  ( V  =  { C ,  A ,  B }  /\  V USGrph  E ) )
2726biimpi 197 . . . . 5  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { C ,  A ,  B }  /\  V USGrph  E
) )
28 nb3graprlem1 25155 . . . . 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 479 . . . 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 271 . . 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 638 . 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 984 . 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 266 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   {cpr 3995   {ctp 3997   <.cop 3999   class class class wbr 4417   ran crn 4847  (class class class)co 6297   USGrph cusg 25034   Neighbors cnbgra 25121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4530  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589  ax-cnex 9591  ax-resscn 9592  ax-1cn 9593  ax-icn 9594  ax-addcl 9595  ax-addrcl 9596  ax-mulcl 9597  ax-mulrcl 9598  ax-mulcom 9599  ax-addass 9600  ax-mulass 9601  ax-distr 9602  ax-i2m1 9603  ax-1ne0 9604  ax-1rid 9605  ax-rnegex 9606  ax-rrecex 9607  ax-cnre 9608  ax-pre-lttri 9609  ax-pre-lttrn 9610  ax-pre-ltadd 9611  ax-pre-mulgt0 9612
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4477  df-mpt 4478  df-tr 4513  df-eprel 4757  df-id 4761  df-po 4767  df-so 4768  df-fr 4805  df-we 4807  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-pred 5391  df-ord 5437  df-on 5438  df-lim 5439  df-suc 5440  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6259  df-ov 6300  df-oprab 6301  df-mpt2 6302  df-om 6699  df-1st 6799  df-2nd 6800  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8370  df-cda 8594  df-pnf 9673  df-mnf 9674  df-xr 9675  df-ltxr 9676  df-le 9677  df-sub 9858  df-neg 9859  df-nn 10606  df-2 10664  df-n0 10866  df-z 10934  df-uz 11156  df-fz 11779  df-hash 12509  df-usgra 25037  df-nbgra 25124
This theorem is referenced by: (None)
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