MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nb3grapr2 Structured version   Unicode version

Theorem nb3grapr2 21455
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 948 . . . . 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 11 . . . 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 3874 . . . . . . . . . . 11  |-  { C ,  A }  =  { A ,  C }
43eleq1i 2498 . . . . . . . . . 10  |-  ( { C ,  A }  e.  ran  E  <->  { A ,  C }  e.  ran  E )
54biimpi 187 . . . . . . . . 9  |-  ( { C ,  A }  e.  ran  E  ->  { A ,  C }  e.  ran  E )
65pm4.71i 614 . . . . . . . 8  |-  ( { C ,  A }  e.  ran  E  <->  ( { C ,  A }  e.  ran  E  /\  { A ,  C }  e.  ran  E ) )
76anbi2i 676 . . . . . . 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 631 . . . . . . 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 244 . . . . . 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 677 . . . . 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 631 . . . . 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 241 . . . 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 253 . . 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 3874 . . . . . . . . . 10  |-  { A ,  B }  =  { B ,  A }
1514eleq1i 2498 . . . . . . . . 9  |-  ( { A ,  B }  e.  ran  E  <->  { B ,  A }  e.  ran  E )
1615biimpi 187 . . . . . . . 8  |-  ( { A ,  B }  e.  ran  E  ->  { B ,  A }  e.  ran  E )
1716pm4.71i 614 . . . . . . 7  |-  ( { A ,  B }  e.  ran  E  <->  ( { A ,  B }  e.  ran  E  /\  { B ,  A }  e.  ran  E ) )
1817anbi1i 677 . . . . . 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 938 . . . . . 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 244 . . . . 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 3874 . . . . . . . . . 10  |-  { B ,  C }  =  { C ,  B }
2221eleq1i 2498 . . . . . . . . 9  |-  ( { B ,  C }  e.  ran  E  <->  { C ,  B }  e.  ran  E )
2322biimpi 187 . . . . . . . 8  |-  ( { B ,  C }  e.  ran  E  ->  { C ,  B }  e.  ran  E )
2423pm4.71i 614 . . . . . . 7  |-  ( { B ,  C }  e.  ran  E  <->  ( { B ,  C }  e.  ran  E  /\  { C ,  B }  e.  ran  E ) )
2524anbi2i 676 . . . . . 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 940 . . . . . 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 244 . . . . 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 679 . . . 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 1263 . . . 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 241 . . 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 253 . 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 21452 . . . . 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 193 . . . 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 943 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  <->  ( B  e.  Y  /\  A  e.  X  /\  C  e.  Z )
)
3534biimpi 187 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( B  e.  Y  /\  A  e.  X  /\  C  e.  Z
) )
36 tpcoma 3892 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  A ,  C }
3736eqeq2i 2445 . . . . . . . 8  |-  ( V  =  { A ,  B ,  C }  <->  V  =  { B ,  A ,  C }
)
3837biimpi 187 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { B ,  A ,  C }
)
3938anim1i 552 . . . . . 6  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { B ,  A ,  C }  /\  V USGrph  E
) )
40 nb3graprlem1 21452 . . . . . 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 464 . . . . 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 193 . . . 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 941 . . . . . . 7  |-  ( ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y )  <->  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )
)
4443biimpri 198 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( C  e.  Z  /\  A  e.  X  /\  B  e.  Y
) )
45 tprot 3891 . . . . . . . . . 10  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
4645eqcomi 2439 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { C ,  A ,  B }
4746eqeq2i 2445 . . . . . . . 8  |-  ( V  =  { A ,  B ,  C }  <->  V  =  { C ,  A ,  B }
)
4847biimpi 187 . . . . . . 7  |-  ( V  =  { A ,  B ,  C }  ->  V  =  { C ,  A ,  B }
)
4948anim1i 552 . . . . . 6  |-  ( ( V  =  { A ,  B ,  C }  /\  V USGrph  E )  -> 
( V  =  { C ,  A ,  B }  /\  V USGrph  E
) )
50 nb3graprlem1 21452 . . . . . 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 464 . . . . 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 193 . . . 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 1254 . . 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 977 . 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 245 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   {cpr 3807   {ctp 3808   <.cop 3809   class class class wbr 4204   ran crn 4871  (class class class)co 6073   USGrph cusg 21357   Neighbors cnbgra 21422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-hash 11611  df-usgra 21359  df-nbgra 21425
  Copyright terms: Public domain W3C validator