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Theorem nb3grapr2 27400
Description: The neighbors of a vertex in a graph with three elements are an unordered pair of the other vertices if and only if all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
Assertion
Ref Expression
nb3grapr2  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( ( <. V ,  E >. Neighbors  A
)  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B
)  =  { A ,  C }  /\  ( <. V ,  E >. Neighbors  C
)  =  { A ,  B } ) ) )

Proof of Theorem nb3grapr2
StepHypRef Expression
1 3anan32 946 . . . . 5  |-  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  { B ,  C }  e.  ran  E ) )
21a1i 10 . . . 4  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  { B ,  C }  e.  ran  E ) ) )
3 prcom 3739 . . . . . . . . . . 11  |-  { C ,  A }  =  { A ,  C }
43eleq1i 2379 . . . . . . . . . 10  |-  ( { C ,  A }  e.  ran  E  <->  { A ,  C }  e.  ran  E )
54biimpi 186 . . . . . . . . 9  |-  ( { C ,  A }  e.  ran  E  ->  { A ,  C }  e.  ran  E )
65pm4.71i 613 . . . . . . . 8  |-  ( { C ,  A }  e.  ran  E  <->  ( { C ,  A }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )
76anbi2i 675 . . . . . . 7  |-  ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( { A ,  B }  e.  ran  E  /\  ( { C ,  A }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )
8 anass 630 . . . . . . 7  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  { A ,  C }  e.  ran  E )  <->  ( { A ,  B }  e.  ran  E  /\  ( { C ,  A }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) ) )
97, 8bitr4i 243 . . . . . 6  |-  ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  { A ,  C }  e.  ran  E ) )
109anbi1i 676 . . . . 5  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  { B ,  C }  e.  ran  E )  <->  ( ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  { A ,  C }  e.  ran  E )  /\  { B ,  C }  e.  ran  E ) )
11 anass 630 . . . . 5  |-  ( ( ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  { A ,  C }  e.  ran  E )  /\  { B ,  C }  e.  ran  E )  <->  ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  ( { A ,  C }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
1210, 11bitri 240 . . . 4  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  { B ,  C }  e.  ran  E )  <->  ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  ( { A ,  C }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
132, 12syl6bb 252 . . 3  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  ( { A ,  C }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
14 prcom 3739 . . . . . . . . . 10  |-  { A ,  B }  =  { B ,  A }
1514eleq1i 2379 . . . . . . . . 9  |-  ( { A ,  B }  e.  ran  E  <->  { B ,  A }  e.  ran  E )
1615biimpi 186 . . . . . . . 8  |-  ( { A ,  B }  e.  ran  E  ->  { B ,  A }  e.  ran  E )
1716pm4.71i 613 . . . . . . 7  |-  ( { A ,  B }  e.  ran  E  <->  ( { A ,  B }  e.  ran  E  /\  { B ,  A }  e.  ran  E ) )
1817anbi1i 676 . . . . . 6  |-  ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  A }  e.  ran  E )  /\  { C ,  A }  e.  ran  E ) )
19 df-3an 936 . . . . . 6  |-  ( ( { A ,  B }  e.  ran  E  /\  { B ,  A }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  A }  e.  ran  E )  /\  { C ,  A }  e.  ran  E ) )
2018, 19bitr4i 243 . . . . 5  |-  ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( { A ,  B }  e.  ran  E  /\  { B ,  A }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )
21 prcom 3739 . . . . . . . . . 10  |-  { B ,  C }  =  { C ,  B }
2221eleq1i 2379 . . . . . . . . 9  |-  ( { B ,  C }  e.  ran  E  <->  { C ,  B }  e.  ran  E )
2322biimpi 186 . . . . . . . 8  |-  ( { B ,  C }  e.  ran  E  ->  { C ,  B }  e.  ran  E )
2423pm4.71i 613 . . . . . . 7  |-  ( { B ,  C }  e.  ran  E  <->  ( { B ,  C }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
2524anbi2i 675 . . . . . 6  |-  ( ( { A ,  C }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  <->  ( { A ,  C }  e.  ran  E  /\  ( { B ,  C }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
26 3anass 938 . . . . . 6  |-  ( ( { A ,  C }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  B }  e.  ran  E )  <->  ( { A ,  C }  e.  ran  E  /\  ( { B ,  C }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
2725, 26bitr4i 243 . . . . 5  |-  ( ( { A ,  C }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  <->  ( { A ,  C }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
2820, 27anbi12i 678 . . . 4  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  ( { A ,  C }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  <-> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  A }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  ( { A ,  C }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) ) )
29 an6 1261 . . . 4  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  A }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  ( { A ,  C }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  B }  e.  ran  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 ) ) )
3028, 29bitri 240 . . 3  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  /\  ( { A ,  C }  e.  ran  E  /\  { B ,  C }  e.  ran  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 ) ) )
3113, 30syl6bb 252 . 2  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  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 ) ) ) )
32 nb3graprlem1 27397 . . . . 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 ) ) )
3332bicomd 192 . . . 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 )  <->  ( <. V ,  E >. Neighbors  A )  =  { B ,  C }
) )
34 3ancoma 941 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  <->  ( B  e.  Y  /\  A  e.  X  /\  C  e.  Z )
)
3534biimpi 186 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( B  e.  Y  /\  A  e.  X  /\  C  e.  Z
) )
36 tpcoma 3757 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  A ,  C }
3736eqeq2i 2326 . . . . . . . 8  |-  ( V  =  { A ,  B ,  C }  <->  V  =  { B ,  A ,  C }
)
3837biimpi 186 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { B ,  A ,  C }
)
3938anim1i 551 . . . . . 6  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { B ,  A ,  C }  /\  V USGrph  E
) )
40 nb3graprlem1 27397 . . . . . 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 ) ) )
4135, 39, 40syl2an 463 . . . . 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 ) ) )
4241bicomd 192 . . . 4  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( { B ,  A }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  <->  ( <. V ,  E >. Neighbors  B )  =  { A ,  C }
) )
43 3anrot 939 . . . . . . 7  |-  ( ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y )  <->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )
)
4443biimpri 197 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y
) )
45 tprot 3756 . . . . . . . . . 10  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
4645eqcomi 2320 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { C ,  A ,  B }
4746eqeq2i 2326 . . . . . . . 8  |-  ( V  =  { A ,  B ,  C }  <->  V  =  { C ,  A ,  B }
)
4847biimpi 186 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { C ,  A ,  B }
)
4948anim1i 551 . . . . . 6  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { C ,  A ,  B }  /\  V USGrph  E
) )
50 nb3graprlem1 27397 . . . . . 6  |-  ( ( ( 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 ) ) )
5144, 49, 50syl2an 463 . . . . 5  |-  ( ( ( 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 ) ) )
5251bicomd 192 . . . 4  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E ) )  ->  (
( { C ,  A }  e.  ran  E  /\  { C ,  B }  e.  ran  E )  <->  ( <. V ,  E >. Neighbors  C )  =  { A ,  B }
) )
5333, 42, 523anbi123d 1252 . . 3  |-  ( ( ( 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 ) )  <-> 
( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B )  =  { A ,  C }  /\  ( <. V ,  E >. Neighbors  C )  =  { A ,  B }
) ) )
54533adant3 975 . 2  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( ( { 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 ) )  <-> 
( ( <. V ,  E >. Neighbors  A )  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B )  =  { A ,  C }  /\  ( <. V ,  E >. Neighbors  C )  =  { A ,  B }
) ) )
5531, 54bitrd 244 1  |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z
)  /\  ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  <->  ( ( <. V ,  E >. Neighbors  A
)  =  { B ,  C }  /\  ( <. V ,  E >. Neighbors  B
)  =  { A ,  C }  /\  ( <. V ,  E >. Neighbors  C
)  =  { A ,  B } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   {cpr 3675   {ctp 3676   <.cop 3677   class class class wbr 4060   ran crn 4727  (class class class)co 5900   USGrph cusg 27317   Neighbors cnbgra 27367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-card 7617  df-cda 7839  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-hash 11385  df-usgra 27319  df-nbgra 27370
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