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Theorem 4cycl2vnunb 30534
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 30533 . . 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 3952 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  B  ->  { A ,  x }  =  { A ,  B }
)
3 preq1 3951 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  B  ->  { x ,  C }  =  { B ,  C }
)
42, 3preq12d 3959 . . . . . . . . . . . . . . . . 17  |-  ( x  =  B  ->  { { A ,  x } ,  { x ,  C } }  =  { { A ,  B } ,  { B ,  C } } )
54sseq1d 3380 . . . . . . . . . . . . . . . 16  |-  ( x  =  B  ->  ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  <->  { { A ,  B } ,  { B ,  C } }  C_  ran  E ) )
65anbi1d 699 . . . . . . . . . . . . . . 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 2614 . . . . . . . . . . . . . . 15  |-  ( x  =  B  ->  (
x  =/=  y  <->  B  =/=  y ) )
86, 7anbi12d 705 . . . . . . . . . . . . . 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 3952 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  D  ->  { A ,  y }  =  { A ,  D }
)
10 preq1 3951 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  D  ->  { y ,  C }  =  { D ,  C }
)
119, 10preq12d 3959 . . . . . . . . . . . . . . . . 17  |-  ( y  =  D  ->  { { A ,  y } ,  { y ,  C } }  =  { { A ,  D } ,  { D ,  C } } )
1211sseq1d 3380 . . . . . . . . . . . . . . . 16  |-  ( y  =  D  ->  ( { { A ,  y } ,  { y ,  C } }  C_ 
ran  E  <->  { { A ,  D } ,  { D ,  C } }  C_  ran  E ) )
1312anbi2d 698 . . . . . . . . . . . . . . 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 2615 . . . . . . . . . . . . . . 15  |-  ( y  =  D  ->  ( B  =/=  y  <->  B  =/=  D ) )
1513, 14anbi12d 705 . . . . . . . . . . . . . 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 3078 . . . . . . . . . . . . 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 1182 . . . . . . . . . . . 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 435 . . . . . . . . . . 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 434 . . . . . . . . . 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 80 . . . . . . . . 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 1179 . . . . . . . 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 430 . . . . . . 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 2724 . . . . . . . 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 2724 . . . . . . . . . 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 425 . . . . . . . . . . . 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 2606 . . . . . . . . . . . . . 14  |-  ( x  =/=  y  <->  -.  x  =  y )
2726bicomi 202 . . . . . . . . . . . . 13  |-  ( -.  x  =  y  <->  x  =/=  y )
2827anbi2i 689 . . . . . . . . . . . 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 251 . . . . . . . . . . 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 2738 . . . . . . . . . 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 251 . . . . . . . . 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 2738 . . . . . . . 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 251 . . . . . . 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 212 . . . . . 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 902 . . . . 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 3952 . . . . . . . 8  |-  ( x  =  y  ->  { A ,  x }  =  { A ,  y }
)
37 preq1 3951 . . . . . . . 8  |-  ( x  =  y  ->  { x ,  C }  =  {
y ,  C }
)
3836, 37preq12d 3959 . . . . . . 7  |-  ( x  =  y  ->  { { A ,  x } ,  { x ,  C } }  =  { { A ,  y } ,  { y ,  C } } )
3938sseq1d 3380 . . . . . 6  |-  ( x  =  y  ->  ( { { A ,  x } ,  { x ,  C } }  C_  ran  E  <->  { { A , 
y } ,  {
y ,  C } }  C_  ran  E ) )
4039reu4 3150 . . . . 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 305 . . . 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 434 . . 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 1179 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   E!wreu 2715    C_ wss 3325   {cpr 3876   ran crn 4837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-v 2972  df-un 3330  df-in 3332  df-ss 3339  df-sn 3875  df-pr 3877
This theorem is referenced by:  n4cyclfrgra  30535  4cyclusnfrgra  30536
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