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Theorem usgvincvadALT 40227
Description: If there is a vertex being incident with an edge in a graph, there is a(nother) vertex in the graph being adjacent to the given vertex by the given edge, analogous to usgraedg4 25193. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 9-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
usgedgimpALT.v  |-  V  =  ( 1st `  G
)
usgedgimpALT.e  |-  E  =  ( Edges  `  G )
Assertion
Ref Expression
usgvincvadALT  |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  E. y  e.  V  C  =  { X ,  y } )
Distinct variable groups:    y, C    y, E    y, G    y, V    y, X

Proof of Theorem usgvincvadALT
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgedgimpALT.v . . . 4  |-  V  =  ( 1st `  G
)
2 usgedgimpALT.e . . . 4  |-  E  =  ( Edges  `  G )
31, 2usgedgimpALT 40226 . . 3  |-  ( ( G  e. USGrph  /\  C  e.  E )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  C  =  { a ,  b } ) )
433adant3 1050 . 2  |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  C  =  { a ,  b } ) )
5 eleq2 2538 . . . . . . . . 9  |-  ( C  =  { a ,  b }  ->  ( X  e.  C  <->  X  e.  { a ,  b } ) )
6 elpri 3976 . . . . . . . . . 10  |-  ( X  e.  { a ,  b }  ->  ( X  =  a  \/  X  =  b )
)
7 andir 885 . . . . . . . . . . . 12  |-  ( ( ( X  =  a  \/  X  =  b )  /\  C  =  { a ,  b } )  <->  ( ( X  =  a  /\  C  =  { a ,  b } )  \/  ( X  =  b  /\  C  =  { a ,  b } ) ) )
87biimpi 199 . . . . . . . . . . 11  |-  ( ( ( X  =  a  \/  X  =  b )  /\  C  =  { a ,  b } )  ->  (
( X  =  a  /\  C  =  {
a ,  b } )  \/  ( X  =  b  /\  C  =  { a ,  b } ) ) )
98ex 441 . . . . . . . . . 10  |-  ( ( X  =  a  \/  X  =  b )  ->  ( C  =  { a ,  b }  ->  ( ( X  =  a  /\  C  =  { a ,  b } )  \/  ( X  =  b  /\  C  =  { a ,  b } ) ) ) )
106, 9syl 17 . . . . . . . . 9  |-  ( X  e.  { a ,  b }  ->  ( C  =  { a ,  b }  ->  ( ( X  =  a  /\  C  =  {
a ,  b } )  \/  ( X  =  b  /\  C  =  { a ,  b } ) ) ) )
115, 10syl6bi 236 . . . . . . . 8  |-  ( C  =  { a ,  b }  ->  ( X  e.  C  ->  ( C  =  { a ,  b }  ->  ( ( X  =  a  /\  C  =  {
a ,  b } )  \/  ( X  =  b  /\  C  =  { a ,  b } ) ) ) ) )
1211pm2.43b 51 . . . . . . 7  |-  ( X  e.  C  ->  ( C  =  { a ,  b }  ->  ( ( X  =  a  /\  C  =  {
a ,  b } )  \/  ( X  =  b  /\  C  =  { a ,  b } ) ) ) )
1312adantld 474 . . . . . 6  |-  ( X  e.  C  ->  (
( a  =/=  b  /\  C  =  {
a ,  b } )  ->  ( ( X  =  a  /\  C  =  { a ,  b } )  \/  ( X  =  b  /\  C  =  { a ,  b } ) ) ) )
14133ad2ant3 1053 . . . . 5  |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  (
( a  =/=  b  /\  C  =  {
a ,  b } )  ->  ( ( X  =  a  /\  C  =  { a ,  b } )  \/  ( X  =  b  /\  C  =  { a ,  b } ) ) ) )
1514reximdv 2857 . . . 4  |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  ( E. b  e.  V  ( a  =/=  b  /\  C  =  {
a ,  b } )  ->  E. b  e.  V  ( ( X  =  a  /\  C  =  { a ,  b } )  \/  ( X  =  b  /\  C  =  { a ,  b } ) ) ) )
1615reximdv 2857 . . 3  |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  ( E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  C  =  {
a ,  b } )  ->  E. a  e.  V  E. b  e.  V  ( ( X  =  a  /\  C  =  { a ,  b } )  \/  ( X  =  b  /\  C  =  { a ,  b } ) ) ) )
17 r19.43 2932 . . . . . 6  |-  ( E. b  e.  V  ( ( X  =  a  /\  C  =  {
a ,  b } )  \/  ( X  =  b  /\  C  =  { a ,  b } ) )  <->  ( E. b  e.  V  ( X  =  a  /\  C  =  { a ,  b } )  \/  E. b  e.  V  ( X  =  b  /\  C  =  { a ,  b } ) ) )
1817rexbii 2881 . . . . 5  |-  ( E. a  e.  V  E. b  e.  V  (
( X  =  a  /\  C  =  {
a ,  b } )  \/  ( X  =  b  /\  C  =  { a ,  b } ) )  <->  E. a  e.  V  ( E. b  e.  V  ( X  =  a  /\  C  =  { a ,  b } )  \/  E. b  e.  V  ( X  =  b  /\  C  =  { a ,  b } ) ) )
19 r19.43 2932 . . . . 5  |-  ( E. a  e.  V  ( E. b  e.  V  ( X  =  a  /\  C  =  {
a ,  b } )  \/  E. b  e.  V  ( X  =  b  /\  C  =  { a ,  b } ) )  <->  ( E. a  e.  V  E. b  e.  V  ( X  =  a  /\  C  =  { a ,  b } )  \/  E. a  e.  V  E. b  e.  V  ( X  =  b  /\  C  =  { a ,  b } ) ) )
2018, 19bitri 257 . . . 4  |-  ( E. a  e.  V  E. b  e.  V  (
( X  =  a  /\  C  =  {
a ,  b } )  \/  ( X  =  b  /\  C  =  { a ,  b } ) )  <->  ( E. a  e.  V  E. b  e.  V  ( X  =  a  /\  C  =  { a ,  b } )  \/  E. a  e.  V  E. b  e.  V  ( X  =  b  /\  C  =  { a ,  b } ) ) )
21 simplr 770 . . . . . . . . 9  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( X  =  a  /\  C  =  { a ,  b } ) )  -> 
b  e.  V )
22 preq2 4043 . . . . . . . . . . 11  |-  ( y  =  b  ->  { X ,  y }  =  { X ,  b } )
2322eqeq2d 2481 . . . . . . . . . 10  |-  ( y  =  b  ->  ( C  =  { X ,  y }  <->  C  =  { X ,  b } ) )
2423adantl 473 . . . . . . . . 9  |-  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  ( X  =  a  /\  C  =  { a ,  b } ) )  /\  y  =  b )  ->  ( C  =  { X ,  y }  <->  C  =  { X ,  b } ) )
25 preq1 4042 . . . . . . . . . . . . 13  |-  ( a  =  X  ->  { a ,  b }  =  { X ,  b } )
2625eqcoms 2479 . . . . . . . . . . . 12  |-  ( X  =  a  ->  { a ,  b }  =  { X ,  b } )
2726eqeq2d 2481 . . . . . . . . . . 11  |-  ( X  =  a  ->  ( C  =  { a ,  b }  <->  C  =  { X ,  b } ) )
2827biimpa 492 . . . . . . . . . 10  |-  ( ( X  =  a  /\  C  =  { a ,  b } )  ->  C  =  { X ,  b }
)
2928adantl 473 . . . . . . . . 9  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( X  =  a  /\  C  =  { a ,  b } ) )  ->  C  =  { X ,  b } )
3021, 24, 29rspcedvd 3143 . . . . . . . 8  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( X  =  a  /\  C  =  { a ,  b } ) )  ->  E. y  e.  V  C  =  { X ,  y } )
3130ex 441 . . . . . . 7  |-  ( ( a  e.  V  /\  b  e.  V )  ->  ( ( X  =  a  /\  C  =  { a ,  b } )  ->  E. y  e.  V  C  =  { X ,  y } ) )
3231rexlimivv 2876 . . . . . 6  |-  ( E. a  e.  V  E. b  e.  V  ( X  =  a  /\  C  =  { a ,  b } )  ->  E. y  e.  V  C  =  { X ,  y } )
33 simpll 768 . . . . . . . . 9  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( X  =  b  /\  C  =  { a ,  b } ) )  -> 
a  e.  V )
34 preq2 4043 . . . . . . . . . . 11  |-  ( y  =  a  ->  { X ,  y }  =  { X ,  a } )
3534eqeq2d 2481 . . . . . . . . . 10  |-  ( y  =  a  ->  ( C  =  { X ,  y }  <->  C  =  { X ,  a } ) )
3635adantl 473 . . . . . . . . 9  |-  ( ( ( ( a  e.  V  /\  b  e.  V )  /\  ( X  =  b  /\  C  =  { a ,  b } ) )  /\  y  =  a )  ->  ( C  =  { X ,  y }  <->  C  =  { X ,  a } ) )
37 preq2 4043 . . . . . . . . . . . . . 14  |-  ( b  =  X  ->  { a ,  b }  =  { a ,  X } )
3837eqcoms 2479 . . . . . . . . . . . . 13  |-  ( X  =  b  ->  { a ,  b }  =  { a ,  X } )
39 prcom 4041 . . . . . . . . . . . . 13  |-  { a ,  X }  =  { X ,  a }
4038, 39syl6eq 2521 . . . . . . . . . . . 12  |-  ( X  =  b  ->  { a ,  b }  =  { X ,  a } )
4140eqeq2d 2481 . . . . . . . . . . 11  |-  ( X  =  b  ->  ( C  =  { a ,  b }  <->  C  =  { X ,  a } ) )
4241biimpa 492 . . . . . . . . . 10  |-  ( ( X  =  b  /\  C  =  { a ,  b } )  ->  C  =  { X ,  a }
)
4342adantl 473 . . . . . . . . 9  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( X  =  b  /\  C  =  { a ,  b } ) )  ->  C  =  { X ,  a } )
4433, 36, 43rspcedvd 3143 . . . . . . . 8  |-  ( ( ( a  e.  V  /\  b  e.  V
)  /\  ( X  =  b  /\  C  =  { a ,  b } ) )  ->  E. y  e.  V  C  =  { X ,  y } )
4544ex 441 . . . . . . 7  |-  ( ( a  e.  V  /\  b  e.  V )  ->  ( ( X  =  b  /\  C  =  { a ,  b } )  ->  E. y  e.  V  C  =  { X ,  y } ) )
4645rexlimivv 2876 . . . . . 6  |-  ( E. a  e.  V  E. b  e.  V  ( X  =  b  /\  C  =  { a ,  b } )  ->  E. y  e.  V  C  =  { X ,  y } )
4732, 46jaoi 386 . . . . 5  |-  ( ( E. a  e.  V  E. b  e.  V  ( X  =  a  /\  C  =  {
a ,  b } )  \/  E. a  e.  V  E. b  e.  V  ( X  =  b  /\  C  =  { a ,  b } ) )  ->  E. y  e.  V  C  =  { X ,  y } )
4847a1i 11 . . . 4  |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  (
( E. a  e.  V  E. b  e.  V  ( X  =  a  /\  C  =  { a ,  b } )  \/  E. a  e.  V  E. b  e.  V  ( X  =  b  /\  C  =  { a ,  b } ) )  ->  E. y  e.  V  C  =  { X ,  y } ) )
4920, 48syl5bi 225 . . 3  |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  ( E. a  e.  V  E. b  e.  V  ( ( X  =  a  /\  C  =  { a ,  b } )  \/  ( X  =  b  /\  C  =  { a ,  b } ) )  ->  E. y  e.  V  C  =  { X ,  y } ) )
5016, 49syld 44 . 2  |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  ( E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  C  =  {
a ,  b } )  ->  E. y  e.  V  C  =  { X ,  y } ) )
514, 50mpd 15 1  |-  ( ( G  e. USGrph  /\  C  e.  E  /\  X  e.  C )  ->  E. y  e.  V  C  =  { X ,  y } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   {cpr 3961   ` cfv 5589   1stc1st 6810   USGrph cusg 25136   Edges cedg 25137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-usgra 25139  df-edg 25142
This theorem is referenced by:  usgvincvadeuALT  40228
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