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Theorem 4cycl2vnunb 27772
Description: In a 4-cycle, two distinct vertices have not a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Nov-2017.)
Assertion
Ref Expression
4cycl2vnunb  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E )  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/=  D
) )  ->  -.  E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E )
Distinct variable groups:    x, A    x, B    x, C    x, E    x, V
Allowed substitution hint:    D( x)

Proof of Theorem 4cycl2vnunb
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 4cycl2v2nb 27771 . . 3  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  ->  ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran  E ) )
2 preq2 3829 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  B  ->  { A ,  x }  =  { A ,  B }
)
3 preq1 3828 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  B  ->  { x ,  C }  =  { B ,  C }
)
42, 3preq12d 3836 . . . . . . . . . . . . . . . . 17  |-  ( x  =  B  ->  { { A ,  x } ,  { x ,  C } }  =  { { A ,  B } ,  { B ,  C } } )
54sseq1d 3320 . . . . . . . . . . . . . . . 16  |-  ( x  =  B  ->  ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  <->  { { A ,  B } ,  { B ,  C } }  C_  ran  E ) )
65anbi1d 686 . . . . . . . . . . . . . . 15  |-  ( x  =  B  ->  (
( { { A ,  x } ,  {
x ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_ 
ran  E )  <->  ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A , 
y } ,  {
y ,  C } }  C_  ran  E ) ) )
7 neeq1 2560 . . . . . . . . . . . . . . 15  |-  ( x  =  B  ->  (
x  =/=  y  <->  B  =/=  y ) )
86, 7anbi12d 692 . . . . . . . . . . . . . 14  |-  ( x  =  B  ->  (
( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A , 
y } ,  {
y ,  C } }  C_  ran  E )  /\  x  =/=  y
)  <->  ( ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_  ran  E
)  /\  B  =/=  y ) ) )
9 preq2 3829 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  D  ->  { A ,  y }  =  { A ,  D }
)
10 preq1 3828 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  D  ->  { y ,  C }  =  { D ,  C }
)
119, 10preq12d 3836 . . . . . . . . . . . . . . . . 17  |-  ( y  =  D  ->  { { A ,  y } ,  { y ,  C } }  =  { { A ,  D } ,  { D ,  C } } )
1211sseq1d 3320 . . . . . . . . . . . . . . . 16  |-  ( y  =  D  ->  ( { { A ,  y } ,  { y ,  C } }  C_ 
ran  E  <->  { { A ,  D } ,  { D ,  C } }  C_  ran  E ) )
1312anbi2d 685 . . . . . . . . . . . . . . 15  |-  ( y  =  D  ->  (
( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_ 
ran  E )  <->  ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran  E ) ) )
14 neeq2 2561 . . . . . . . . . . . . . . 15  |-  ( y  =  D  ->  ( B  =/=  y  <->  B  =/=  D ) )
1513, 14anbi12d 692 . . . . . . . . . . . . . 14  |-  ( y  =  D  ->  (
( ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A , 
y } ,  {
y ,  C } }  C_  ran  E )  /\  B  =/=  y
)  <->  ( ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran  E
)  /\  B  =/=  D ) ) )
168, 15rspc2ev 3005 . . . . . . . . . . . . 13  |-  ( ( B  e.  V  /\  D  e.  V  /\  ( ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran  E )  /\  B  =/=  D ) )  ->  E. x  e.  V  E. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A , 
y } ,  {
y ,  C } }  C_  ran  E )  /\  x  =/=  y
) )
17163expa 1153 . . . . . . . . . . . 12  |-  ( ( ( B  e.  V  /\  D  e.  V
)  /\  ( ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran  E
)  /\  B  =/=  D ) )  ->  E. x  e.  V  E. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_  ran  E
)  /\  x  =/=  y ) )
1817expcom 425 . . . . . . . . . . 11  |-  ( ( ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran  E )  /\  B  =/=  D )  ->  (
( B  e.  V  /\  D  e.  V
)  ->  E. x  e.  V  E. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_  ran  E
)  /\  x  =/=  y ) ) )
1918ex 424 . . . . . . . . . 10  |-  ( ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran  E
)  ->  ( B  =/=  D  ->  ( ( B  e.  V  /\  D  e.  V )  ->  E. x  e.  V  E. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A , 
y } ,  {
y ,  C } }  C_  ran  E )  /\  x  =/=  y
) ) ) )
2019com13 76 . . . . . . . . 9  |-  ( ( B  e.  V  /\  D  e.  V )  ->  ( B  =/=  D  ->  ( ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran  E )  ->  E. x  e.  V  E. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_  ran  E
)  /\  x  =/=  y ) ) ) )
21203impia 1150 . . . . . . . 8  |-  ( ( B  e.  V  /\  D  e.  V  /\  B  =/=  D )  -> 
( ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran  E )  ->  E. x  e.  V  E. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_  ran  E
)  /\  x  =/=  y ) ) )
2221impcom 420 . . . . . . 7  |-  ( ( ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran  E )  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/=  D ) )  ->  E. x  e.  V  E. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A , 
y } ,  {
y ,  C } }  C_  ran  E )  /\  x  =/=  y
) )
23 rexnal 2662 . . . . . . . 8  |-  ( E. x  e.  V  -.  A. y  e.  V  ( ( { { A ,  x } ,  {
x ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_ 
ran  E )  ->  x  =  y )  <->  -. 
A. x  e.  V  A. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A , 
y } ,  {
y ,  C } }  C_  ran  E )  ->  x  =  y ) )
24 rexnal 2662 . . . . . . . . . 10  |-  ( E. y  e.  V  -.  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A , 
y } ,  {
y ,  C } }  C_  ran  E )  ->  x  =  y )  <->  -.  A. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_  ran  E
)  ->  x  =  y ) )
25 annim 415 . . . . . . . . . . . 12  |-  ( ( ( { { A ,  x } ,  {
x ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_ 
ran  E )  /\  -.  x  =  y
)  <->  -.  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_  ran  E
)  ->  x  =  y ) )
26 df-ne 2554 . . . . . . . . . . . . . 14  |-  ( x  =/=  y  <->  -.  x  =  y )
2726bicomi 194 . . . . . . . . . . . . 13  |-  ( -.  x  =  y  <->  x  =/=  y )
2827anbi2i 676 . . . . . . . . . . . 12  |-  ( ( ( { { A ,  x } ,  {
x ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_ 
ran  E )  /\  -.  x  =  y
)  <->  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_  ran  E
)  /\  x  =/=  y ) )
2925, 28bitr3i 243 . . . . . . . . . . 11  |-  ( -.  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A , 
y } ,  {
y ,  C } }  C_  ran  E )  ->  x  =  y )  <->  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_  ran  E
)  /\  x  =/=  y ) )
3029rexbii 2676 . . . . . . . . . 10  |-  ( E. y  e.  V  -.  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A , 
y } ,  {
y ,  C } }  C_  ran  E )  ->  x  =  y )  <->  E. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A , 
y } ,  {
y ,  C } }  C_  ran  E )  /\  x  =/=  y
) )
3124, 30bitr3i 243 . . . . . . . . 9  |-  ( -. 
A. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A , 
y } ,  {
y ,  C } }  C_  ran  E )  ->  x  =  y )  <->  E. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A , 
y } ,  {
y ,  C } }  C_  ran  E )  /\  x  =/=  y
) )
3231rexbii 2676 . . . . . . . 8  |-  ( E. x  e.  V  -.  A. y  e.  V  ( ( { { A ,  x } ,  {
x ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_ 
ran  E )  ->  x  =  y )  <->  E. x  e.  V  E. y  e.  V  (
( { { A ,  x } ,  {
x ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_ 
ran  E )  /\  x  =/=  y ) )
3323, 32bitr3i 243 . . . . . . 7  |-  ( -. 
A. x  e.  V  A. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A , 
y } ,  {
y ,  C } }  C_  ran  E )  ->  x  =  y )  <->  E. x  e.  V  E. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A , 
y } ,  {
y ,  C } }  C_  ran  E )  /\  x  =/=  y
) )
3422, 33sylibr 204 . . . . . 6  |-  ( ( ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran  E )  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/=  D ) )  ->  -.  A. x  e.  V  A. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_  ran  E
)  ->  x  =  y ) )
3534intnand 883 . . . . 5  |-  ( ( ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran  E )  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/=  D ) )  ->  -.  ( E. x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  A. x  e.  V  A. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A , 
y } ,  {
y ,  C } }  C_  ran  E )  ->  x  =  y ) ) )
36 preq2 3829 . . . . . . . 8  |-  ( x  =  y  ->  { A ,  x }  =  { A ,  y }
)
37 preq1 3828 . . . . . . . 8  |-  ( x  =  y  ->  { x ,  C }  =  {
y ,  C }
)
3836, 37preq12d 3836 . . . . . . 7  |-  ( x  =  y  ->  { { A ,  x } ,  { x ,  C } }  =  { { A ,  y } ,  { y ,  C } } )
3938sseq1d 3320 . . . . . 6  |-  ( x  =  y  ->  ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  <->  { { A , 
y } ,  {
y ,  C } }  C_  ran  E ) )
4039reu4 3073 . . . . 5  |-  ( E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E  <->  ( E. x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  A. x  e.  V  A. y  e.  V  ( ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  /\  { { A ,  y } ,  { y ,  C } }  C_  ran  E
)  ->  x  =  y ) ) )
4135, 40sylnibr 297 . . . 4  |-  ( ( ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran  E )  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/=  D ) )  ->  -.  E! x  e.  V  { { A ,  x } ,  {
x ,  C } }  C_  ran  E )
4241ex 424 . . 3  |-  ( ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran  E
)  ->  ( ( B  e.  V  /\  D  e.  V  /\  B  =/=  D )  ->  -.  E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E ) )
431, 42syl 16 . 2  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) )  ->  ( ( B  e.  V  /\  D  e.  V  /\  B  =/= 
D )  ->  -.  E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E ) )
44433impia 1150 1  |-  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E )  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/=  D
) )  ->  -.  E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   E.wrex 2652   E!wreu 2653    C_ wss 3265   {cpr 3760   ran crn 4821
This theorem is referenced by:  n4cyclfrgra  27773  4cyclusnfrgra  27774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-v 2903  df-un 3270  df-in 3272  df-ss 3279  df-sn 3765  df-pr 3766
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