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Theorem usg2spthonot1 30549
Description: A simple path of length 2 between two vertices as ordered triple corresponds to two adjacent edges in an undirected simple graph. (Contributed by Alexander van der Vekens, 9-Mar-2018.)
Assertion
Ref Expression
usg2spthonot1  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V )
)  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( ( T  = 
<. A ,  b ,  C >.  /\  A  =/= 
C )  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) )
Distinct variable groups:    A, b    C, b    E, b    T, b    V, b

Proof of Theorem usg2spthonot1
StepHypRef Expression
1 usgrav 23407 . . 3  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 el2spthonot0 30530 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( T  e.  ( A
( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) ) ) )
31, 2sylan 471 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V )
)  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) ) ) )
4 simpll 753 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  V USGrph  E )
5 simplrl 759 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  A  e.  V )
6 simpr 461 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  b  e.  V )
7 simplrr 760 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  C  e.  V )
8 usg2spthonot0 30548 . . . . . . 7  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  b  e.  V  /\  C  e.  V )
)  ->  ( <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C )  <-> 
( ( A  =  A  /\  C  =  C  /\  A  =/= 
C )  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) )
94, 5, 6, 7, 8syl13anc 1221 . . . . . 6  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  ( <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C )  <->  ( ( A  =  A  /\  C  =  C  /\  A  =/=  C )  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) )
10 simp3 990 . . . . . . . 8  |-  ( ( A  =  A  /\  C  =  C  /\  A  =/=  C )  ->  A  =/=  C )
1110anim1i 568 . . . . . . 7  |-  ( ( ( A  =  A  /\  C  =  C  /\  A  =/=  C
)  /\  ( { A ,  b }  e.  ran  E  /\  {
b ,  C }  e.  ran  E ) )  ->  ( A  =/= 
C  /\  ( { A ,  b }  e.  ran  E  /\  {
b ,  C }  e.  ran  E ) ) )
12 eqidd 2452 . . . . . . . . 9  |-  ( A  =/=  C  ->  A  =  A )
13 eqidd 2452 . . . . . . . . 9  |-  ( A  =/=  C  ->  C  =  C )
14 id 22 . . . . . . . . 9  |-  ( A  =/=  C  ->  A  =/=  C )
1512, 13, 143jca 1168 . . . . . . . 8  |-  ( A  =/=  C  ->  ( A  =  A  /\  C  =  C  /\  A  =/=  C ) )
1615anim1i 568 . . . . . . 7  |-  ( ( A  =/=  C  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) )  ->  ( ( A  =  A  /\  C  =  C  /\  A  =/=  C )  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) )
1711, 16impbii 188 . . . . . 6  |-  ( ( ( A  =  A  /\  C  =  C  /\  A  =/=  C
)  /\  ( { A ,  b }  e.  ran  E  /\  {
b ,  C }  e.  ran  E ) )  <-> 
( A  =/=  C  /\  ( { A , 
b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) )
189, 17syl6bb 261 . . . . 5  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  ( <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C )  <->  ( A  =/=  C  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) )
1918anbi2d 703 . . . 4  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  (
( T  =  <. A ,  b ,  C >.  /\  <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) )  <->  ( T  =  <. A ,  b ,  C >.  /\  ( A  =/=  C  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) ) )
20 anass 649 . . . 4  |-  ( ( ( T  =  <. A ,  b ,  C >.  /\  A  =/=  C
)  /\  ( { A ,  b }  e.  ran  E  /\  {
b ,  C }  e.  ran  E ) )  <-> 
( T  =  <. A ,  b ,  C >.  /\  ( A  =/= 
C  /\  ( { A ,  b }  e.  ran  E  /\  {
b ,  C }  e.  ran  E ) ) ) )
2119, 20syl6bbr 263 . . 3  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  (
( T  =  <. A ,  b ,  C >.  /\  <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) )  <->  ( ( T  =  <. A , 
b ,  C >.  /\  A  =/=  C )  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) )
2221rexbidva 2845 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V )
)  ->  ( E. b  e.  V  ( T  =  <. A , 
b ,  C >.  /\ 
<. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) )  <->  E. b  e.  V  ( ( T  =  <. A , 
b ,  C >.  /\  A  =/=  C )  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) )
233, 22bitrd 253 1  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V )
)  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( ( T  = 
<. A ,  b ,  C >.  /\  A  =/= 
C )  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796   _Vcvv 3070   {cpr 3979   <.cotp 3985   class class class wbr 4392   ran crn 4941  (class class class)co 6192   USGrph cusg 23401   2SPathOnOt c2pthonot 30516
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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-ot 3986  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-pm 7319  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-card 8212  df-cda 8440  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-fzo 11652  df-hash 12207  df-word 12333  df-usgra 23403  df-wlk 23552  df-trail 23553  df-pth 23554  df-spth 23555  df-wlkon 23558  df-spthon 23561  df-2wlkonot 30517  df-2spthonot 30519
This theorem is referenced by:  usg2spot2nb  30798
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