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Theorem usgvad2edg 37973
Description: If a vertex is adjacent to two different vertices in a simple graph, there are more than one edges starting at this vertex, analogous to usgra2edg 24681. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.)
Hypothesis
Ref Expression
usgvad2edg.e  |-  E  =  ( Edges  `  G )
Assertion
Ref Expression
usgvad2edg  |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E
) )  ->  E. x  e.  E  E. y  e.  E  ( x  =/=  y  /\  N  e.  x  /\  N  e.  y ) )
Distinct variable groups:    x, B, y    x, C, y    x, E, y    x, G, y   
x, N, y

Proof of Theorem usgvad2edg
StepHypRef Expression
1 simprl 755 . 2  |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E
) )  ->  { N ,  B }  e.  E
)
2 simprr 756 . 2  |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E
) )  ->  { C ,  N }  e.  E
)
3 usgvad2edg.e . . . . . . . 8  |-  E  =  ( Edges  `  G )
4 eqid 2400 . . . . . . . 8  |-  ( Vtx  `  G )  =  ( Vtx  `  G )
53, 4usgpredgv 37961 . . . . . . 7  |-  ( ( G  e. USGrph  /\  { N ,  B }  e.  E
)  ->  ( N  e.  ( Vtx  `  G )  /\  B  e.  ( Vtx  `  G ) ) )
65ex 432 . . . . . 6  |-  ( G  e. USGrph  ->  ( { N ,  B }  e.  E  ->  ( N  e.  ( Vtx  `  G )  /\  B  e.  ( Vtx  `  G ) ) ) )
73, 4usgpredgv 37961 . . . . . . 7  |-  ( ( G  e. USGrph  /\  { C ,  N }  e.  E
)  ->  ( C  e.  ( Vtx  `  G )  /\  N  e.  ( Vtx  `  G ) ) )
87ex 432 . . . . . 6  |-  ( G  e. USGrph  ->  ( { C ,  N }  e.  E  ->  ( C  e.  ( Vtx  `  G )  /\  N  e.  ( Vtx  `  G ) ) ) )
96, 8anim12d 561 . . . . 5  |-  ( G  e. USGrph  ->  ( ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E
)  ->  ( ( N  e.  ( Vtx  `  G
)  /\  B  e.  ( Vtx  `  G ) )  /\  ( C  e.  ( Vtx  `  G )  /\  N  e.  ( Vtx  `  G ) ) ) ) )
109adantr 463 . . . 4  |-  ( ( G  e. USGrph  /\  B  =/= 
C )  ->  (
( { N ,  B }  e.  E  /\  { C ,  N }  e.  E )  ->  ( ( N  e.  ( Vtx  `  G )  /\  B  e.  ( Vtx  `  G ) )  /\  ( C  e.  ( Vtx  `  G )  /\  N  e.  ( Vtx  `  G ) ) ) ) )
1110imp 427 . . 3  |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E
) )  ->  (
( N  e.  ( Vtx  `  G )  /\  B  e.  ( Vtx  `  G ) )  /\  ( C  e.  ( Vtx  `  G
)  /\  N  e.  ( Vtx  `  G ) ) ) )
12 simplr 754 . . . . 5  |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E
) )  ->  B  =/=  C )
133usgpredgdv 37971 . . . . . . 7  |-  ( ( G  e. USGrph  /\  { N ,  B }  e.  E
)  ->  N  =/=  B )
1413necomd 2672 . . . . . 6  |-  ( ( G  e. USGrph  /\  { N ,  B }  e.  E
)  ->  B  =/=  N )
1514ad2ant2r 745 . . . . 5  |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E
) )  ->  B  =/=  N )
1612, 15jca 530 . . . 4  |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E
) )  ->  ( B  =/=  C  /\  B  =/=  N ) )
1716olcd 391 . . 3  |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E
) )  ->  (
( N  =/=  C  /\  N  =/=  N
)  \/  ( B  =/=  C  /\  B  =/=  N ) ) )
18 prneimg 4150 . . . . 5  |-  ( ( ( N  e.  ( Vtx  `  G )  /\  B  e.  ( Vtx  `  G ) )  /\  ( C  e.  ( Vtx  `  G
)  /\  N  e.  ( Vtx  `  G ) ) )  ->  ( (
( N  =/=  C  /\  N  =/=  N
)  \/  ( B  =/=  C  /\  B  =/=  N ) )  ->  { N ,  B }  =/=  { C ,  N } ) )
1918imp 427 . . . 4  |-  ( ( ( ( N  e.  ( Vtx  `  G )  /\  B  e.  ( Vtx  `  G ) )  /\  ( C  e.  ( Vtx  `  G )  /\  N  e.  ( Vtx  `  G ) ) )  /\  (
( N  =/=  C  /\  N  =/=  N
)  \/  ( B  =/=  C  /\  B  =/=  N ) ) )  ->  { N ,  B }  =/=  { C ,  N } )
20 prid1g 4075 . . . . 5  |-  ( N  e.  ( Vtx  `  G
)  ->  N  e.  { N ,  B }
)
2120ad3antrrr 728 . . . 4  |-  ( ( ( ( N  e.  ( Vtx  `  G )  /\  B  e.  ( Vtx  `  G ) )  /\  ( C  e.  ( Vtx  `  G )  /\  N  e.  ( Vtx  `  G ) ) )  /\  (
( N  =/=  C  /\  N  =/=  N
)  \/  ( B  =/=  C  /\  B  =/=  N ) ) )  ->  N  e.  { N ,  B }
)
22 prid2g 4076 . . . . 5  |-  ( N  e.  ( Vtx  `  G
)  ->  N  e.  { C ,  N }
)
2322ad3antrrr 728 . . . 4  |-  ( ( ( ( N  e.  ( Vtx  `  G )  /\  B  e.  ( Vtx  `  G ) )  /\  ( C  e.  ( Vtx  `  G )  /\  N  e.  ( Vtx  `  G ) ) )  /\  (
( N  =/=  C  /\  N  =/=  N
)  \/  ( B  =/=  C  /\  B  =/=  N ) ) )  ->  N  e.  { C ,  N }
)
2419, 21, 233jca 1175 . . 3  |-  ( ( ( ( N  e.  ( Vtx  `  G )  /\  B  e.  ( Vtx  `  G ) )  /\  ( C  e.  ( Vtx  `  G )  /\  N  e.  ( Vtx  `  G ) ) )  /\  (
( N  =/=  C  /\  N  =/=  N
)  \/  ( B  =/=  C  /\  B  =/=  N ) ) )  ->  ( { N ,  B }  =/=  { C ,  N }  /\  N  e.  { N ,  B }  /\  N  e.  { C ,  N } ) )
2511, 17, 24syl2anc 659 . 2  |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E
) )  ->  ( { N ,  B }  =/=  { C ,  N }  /\  N  e.  { N ,  B }  /\  N  e.  { C ,  N } ) )
26 neeq1 2682 . . . 4  |-  ( x  =  { N ,  B }  ->  ( x  =/=  y  <->  { N ,  B }  =/=  y
) )
27 eleq2 2473 . . . 4  |-  ( x  =  { N ,  B }  ->  ( N  e.  x  <->  N  e.  { N ,  B }
) )
2826, 273anbi12d 1300 . . 3  |-  ( x  =  { N ,  B }  ->  ( ( x  =/=  y  /\  N  e.  x  /\  N  e.  y )  <->  ( { N ,  B }  =/=  y  /\  N  e.  { N ,  B }  /\  N  e.  y ) ) )
29 neeq2 2684 . . . 4  |-  ( y  =  { C ,  N }  ->  ( { N ,  B }  =/=  y  <->  { N ,  B }  =/=  { C ,  N } ) )
30 eleq2 2473 . . . 4  |-  ( y  =  { C ,  N }  ->  ( N  e.  y  <->  N  e.  { C ,  N }
) )
3129, 303anbi13d 1301 . . 3  |-  ( y  =  { C ,  N }  ->  ( ( { N ,  B }  =/=  y  /\  N  e.  { N ,  B }  /\  N  e.  y )  <->  ( { N ,  B }  =/=  { C ,  N }  /\  N  e.  { N ,  B }  /\  N  e.  { C ,  N } ) ) )
3228, 31rspc2ev 3168 . 2  |-  ( ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E  /\  ( { N ,  B }  =/=  { C ,  N }  /\  N  e.  { N ,  B }  /\  N  e.  { C ,  N } ) )  ->  E. x  e.  E  E. y  e.  E  ( x  =/=  y  /\  N  e.  x  /\  N  e.  y
) )
331, 2, 25, 32syl3anc 1228 1  |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E
) )  ->  E. x  e.  E  E. y  e.  E  ( x  =/=  y  /\  N  e.  x  /\  N  e.  y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   E.wrex 2752   {cpr 3971   ` cfv 5523   USGrph cusg 24629   Edges cedg 24630   Vtx cvtx 37943
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-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-hash 12358  df-usgra 24632  df-edg 24635  df-vtx 37944
This theorem is referenced by:  usg2edgneu  37974
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