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Theorem usgra2adedgspth 24912
Description: In an undirected simple graph, two adjacent edges form a simple path of length 2. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
Hypotheses
Ref Expression
usgra2adedgspth.f  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. }
usgra2adedgspth.p  |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }
Assertion
Ref Expression
usgra2adedgspth  |-  ( ( V USGrph  E  /\  A  =/= 
C )  ->  (
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  F ( V SPaths  E ) P ) )

Proof of Theorem usgra2adedgspth
StepHypRef Expression
1 usgrav 24637 . . . 4  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
21ad2antrr 724 . . 3  |-  ( ( ( V USGrph  E  /\  A  =/=  C )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( V  e. 
_V  /\  E  e.  _V ) )
3 usgraedgrnv 24676 . . . . . . 7  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  ( A  e.  V  /\  B  e.  V ) )
43adantrr 715 . . . . . 6  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( A  e.  V  /\  B  e.  V ) )
54simpld 457 . . . . 5  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  A  e.  V
)
64simprd 461 . . . . 5  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  B  e.  V
)
7 usgraedgrnv 24676 . . . . . . 7  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  ( B  e.  V  /\  C  e.  V ) )
87adantrl 714 . . . . . 6  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( B  e.  V  /\  C  e.  V ) )
98simprd 461 . . . . 5  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  C  e.  V
)
105, 6, 93jca 1175 . . . 4  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )
1110adantlr 713 . . 3  |-  ( ( ( V USGrph  E  /\  A  =/=  C )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )
12 usgraedgrn 24680 . . . . 5  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  A  =/=  B )
1312ad2ant2r 745 . . . 4  |-  ( ( ( V USGrph  E  /\  A  =/=  C )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  A  =/=  B
)
14 simplr 754 . . . 4  |-  ( ( ( V USGrph  E  /\  A  =/=  C )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  A  =/=  C
)
15 usgraedgrn 24680 . . . . 5  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  B  =/=  C )
1615ad2ant2rl 747 . . . 4  |-  ( ( ( V USGrph  E  /\  A  =/=  C )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  B  =/=  C
)
1713, 14, 163jca 1175 . . 3  |-  ( ( ( V USGrph  E  /\  A  =/=  C )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( A  =/= 
B  /\  A  =/=  C  /\  B  =/=  C
) )
18 usgra2adedgspthlem2 24911 . . 3  |-  ( ( ( V USGrph  E  /\  A  =/=  C )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( ( `' E `  { A ,  B } )  =/=  ( `' E `  { B ,  C }
)  /\  ( E `  ( `' E `  { A ,  B }
) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } ) )
19 fvex 5813 . . . . . 6  |-  ( `' E `  { A ,  B } )  e. 
_V
20 fvex 5813 . . . . . 6  |-  ( `' E `  { B ,  C } )  e. 
_V
2119, 20pm3.2i 453 . . . . 5  |-  ( ( `' E `  { A ,  B } )  e. 
_V  /\  ( `' E `  { B ,  C } )  e. 
_V )
22 usgra2adedgspth.f . . . . 5  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. }
23 usgra2adedgspth.p . . . . 5  |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }
2421, 22, 23constr2spth 24901 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  A  =/= 
C  /\  B  =/=  C ) )  ->  (
( ( `' E `  { A ,  B } )  =/=  ( `' E `  { B ,  C } )  /\  ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } )  ->  F
( V SPaths  E ) P ) )
2524imp 427 . . 3  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
( `' E `  { A ,  B }
)  =/=  ( `' E `  { B ,  C } )  /\  ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } ) )  ->  F ( V SPaths  E
) P )
262, 11, 17, 18, 25syl31anc 1231 . 2  |-  ( ( ( V USGrph  E  /\  A  =/=  C )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  F ( V SPaths  E ) P )
2726ex 432 1  |-  ( ( V USGrph  E  /\  A  =/= 
C )  ->  (
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  F ( V SPaths  E ) P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   _Vcvv 3056   {cpr 3971   {ctp 3973   <.cop 3975   class class class wbr 4392   `'ccnv 4939   ran crn 4941   ` cfv 5523  (class class class)co 6232   0cc0 9440   1c1 9441   2c2 10544   USGrph cusg 24629   SPaths cspath 24800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  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 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-map 7377  df-pm 7378  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-n0 10755  df-z 10824  df-uz 11044  df-fz 11642  df-fzo 11766  df-hash 12358  df-word 12496  df-usgra 24632  df-wlk 24807  df-trail 24808  df-spth 24810
This theorem is referenced by: (None)
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