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Theorem usgvad2edg 32365
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 24360. (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 756 . 2  |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E
) )  ->  { N ,  B }  e.  E
)
2 simprr 757 . 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 2443 . . . . . . . 8  |-  ( Vtx  `  G )  =  ( Vtx  `  G )
53, 4usgpredgv 32353 . . . . . . 7  |-  ( ( G  e. USGrph  /\  { N ,  B }  e.  E
)  ->  ( N  e.  ( Vtx  `  G )  /\  B  e.  ( Vtx  `  G ) ) )
65ex 434 . . . . . 6  |-  ( G  e. USGrph  ->  ( { N ,  B }  e.  E  ->  ( N  e.  ( Vtx  `  G )  /\  B  e.  ( Vtx  `  G ) ) ) )
73, 4usgpredgv 32353 . . . . . . 7  |-  ( ( G  e. USGrph  /\  { C ,  N }  e.  E
)  ->  ( C  e.  ( Vtx  `  G )  /\  N  e.  ( Vtx  `  G ) ) )
87ex 434 . . . . . 6  |-  ( G  e. USGrph  ->  ( { C ,  N }  e.  E  ->  ( C  e.  ( Vtx  `  G )  /\  N  e.  ( Vtx  `  G ) ) ) )
96, 8anim12d 563 . . . . 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 465 . . . 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 429 . . 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 755 . . . . 5  |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E
) )  ->  B  =/=  C )
133usgpredgdv 32363 . . . . . . 7  |-  ( ( G  e. USGrph  /\  { N ,  B }  e.  E
)  ->  N  =/=  B )
1413necomd 2714 . . . . . 6  |-  ( ( G  e. USGrph  /\  { N ,  B }  e.  E
)  ->  B  =/=  N )
1514ad2ant2r 746 . . . . 5  |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E
) )  ->  B  =/=  N )
1612, 15jca 532 . . . 4  |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E
) )  ->  ( B  =/=  C  /\  B  =/=  N ) )
1716olcd 393 . . 3  |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E
) )  ->  (
( N  =/=  C  /\  N  =/=  N
)  \/  ( B  =/=  C  /\  B  =/=  N ) ) )
18 prneimg 4196 . . . . 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 429 . . . 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 4121 . . . . 5  |-  ( N  e.  ( Vtx  `  G
)  ->  N  e.  { N ,  B }
)
2120ad3antrrr 729 . . . 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 4122 . . . . 5  |-  ( N  e.  ( Vtx  `  G
)  ->  N  e.  { C ,  N }
)
2322ad3antrrr 729 . . . 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 1177 . . 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 661 . 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 2724 . . . 4  |-  ( x  =  { N ,  B }  ->  ( x  =/=  y  <->  { N ,  B }  =/=  y
) )
27 eleq2 2516 . . . 4  |-  ( x  =  { N ,  B }  ->  ( N  e.  x  <->  N  e.  { N ,  B }
) )
2826, 273anbi12d 1301 . . 3  |-  ( x  =  { N ,  B }  ->  ( ( x  =/=  y  /\  N  e.  x  /\  N  e.  y )  <->  ( { N ,  B }  =/=  y  /\  N  e.  { N ,  B }  /\  N  e.  y ) ) )
29 neeq2 2726 . . . 4  |-  ( y  =  { C ,  N }  ->  ( { N ,  B }  =/=  y  <->  { N ,  B }  =/=  { C ,  N } ) )
30 eleq2 2516 . . . 4  |-  ( y  =  { C ,  N }  ->  ( N  e.  y  <->  N  e.  { C ,  N }
) )
3129, 303anbi13d 1302 . . 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 3207 . 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 1229 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 368    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   E.wrex 2794   {cpr 4016   ` cfv 5578   USGrph cusg 24308   Edges cedg 24309   Vtx cvtx 32335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-hash 12388  df-usgra 24311  df-edg 24314  df-vtx 32336
This theorem is referenced by:  usg2edgneu  32366
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