Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frgrawopreg Structured version   Unicode version

Theorem frgrawopreg 30780
Description: In a friendship graph there are either no vertices or exactly one vertex having degree K, or all or all except one vertices have degree K. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
Hypotheses
Ref Expression
frgrawopreg.a  |-  A  =  { x  e.  V  |  ( ( V VDeg 
E ) `  x
)  =  K }
frgrawopreg.b  |-  B  =  ( V  \  A
)
Assertion
Ref Expression
frgrawopreg  |-  ( V FriendGrph  E  ->  ( ( (
# `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) )
Distinct variable groups:    x, A    x, E    x, K    x, V    x, B

Proof of Theorem frgrawopreg
Dummy variables  b 
y  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrawopreg.a . . 3  |-  A  =  { x  e.  V  |  ( ( V VDeg 
E ) `  x
)  =  K }
2 frgrawopreg.b . . 3  |-  B  =  ( V  \  A
)
31, 2frgrawopreglem1 30775 . 2  |-  ( V FriendGrph  E  ->  ( A  e. 
_V  /\  B  e.  _V ) )
4 hashv01gt1 12217 . . . 4  |-  ( A  e.  _V  ->  (
( # `  A )  =  0  \/  ( # `
 A )  =  1  \/  1  < 
( # `  A ) ) )
5 hashv01gt1 12217 . . . 4  |-  ( B  e.  _V  ->  (
( # `  B )  =  0  \/  ( # `
 B )  =  1  \/  1  < 
( # `  B ) ) )
64, 5anim12i 566 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( ( # `  A )  =  0  \/  ( # `  A
)  =  1  \/  1  <  ( # `  A ) )  /\  ( ( # `  B
)  =  0  \/  ( # `  B
)  =  1  \/  1  <  ( # `  B ) ) ) )
7 hasheq0 12232 . . . . . . . . . . . . 13  |-  ( A  e.  _V  ->  (
( # `  A )  =  0  <->  A  =  (/) ) )
87biimpd 207 . . . . . . . . . . . 12  |-  ( A  e.  _V  ->  (
( # `  A )  =  0  ->  A  =  (/) ) )
98adantr 465 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( # `  A
)  =  0  ->  A  =  (/) ) )
109impcom 430 . . . . . . . . . 10  |-  ( ( ( # `  A
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  A  =  (/) )
1110olcd 393 . . . . . . . . 9  |-  ( ( ( # `  A
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( ( # `
 A )  =  1  \/  A  =  (/) ) )
1211orcd 392 . . . . . . . 8  |-  ( ( ( # `  A
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) )
1312a1d 25 . . . . . . 7  |-  ( ( ( # `  A
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( V FriendGrph  E  ->  ( ( (
# `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) )
1413ex 434 . . . . . 6  |-  ( (
# `  A )  =  0  ->  (
( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( ( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) ) ) )
1514a1d 25 . . . . 5  |-  ( (
# `  A )  =  0  ->  (
( ( # `  B
)  =  0  \/  ( # `  B
)  =  1  \/  1  <  ( # `  B ) )  -> 
( ( A  e. 
_V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( ( (
# `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) ) ) )
16 orc 385 . . . . . . . . 9  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  =  1  \/  A  =  (/) ) )
1716orcd 392 . . . . . . . 8  |-  ( (
# `  A )  =  1  ->  (
( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) )
1817a1d 25 . . . . . . 7  |-  ( (
# `  A )  =  1  ->  ( V FriendGrph  E  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) )
1918a1d 25 . . . . . 6  |-  ( (
# `  A )  =  1  ->  (
( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( ( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) ) ) )
2019a1d 25 . . . . 5  |-  ( (
# `  A )  =  1  ->  (
( ( # `  B
)  =  0  \/  ( # `  B
)  =  1  \/  1  <  ( # `  B ) )  -> 
( ( A  e. 
_V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( ( (
# `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) ) ) )
21 hasheq0 12232 . . . . . . . . . . . . . . 15  |-  ( B  e.  _V  ->  (
( # `  B )  =  0  <->  B  =  (/) ) )
2221biimpd 207 . . . . . . . . . . . . . 14  |-  ( B  e.  _V  ->  (
( # `  B )  =  0  ->  B  =  (/) ) )
2322adantl 466 . . . . . . . . . . . . 13  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( # `  B
)  =  0  ->  B  =  (/) ) )
2423impcom 430 . . . . . . . . . . . 12  |-  ( ( ( # `  B
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  B  =  (/) )
2524olcd 393 . . . . . . . . . . 11  |-  ( ( ( # `  B
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( ( # `
 B )  =  1  \/  B  =  (/) ) )
2625olcd 393 . . . . . . . . . 10  |-  ( ( ( # `  B
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) )
2726a1d 25 . . . . . . . . 9  |-  ( ( ( # `  B
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( V FriendGrph  E  ->  ( ( (
# `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) )
2827ex 434 . . . . . . . 8  |-  ( (
# `  B )  =  0  ->  (
( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( ( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) ) ) )
2928a1d 25 . . . . . . 7  |-  ( (
# `  B )  =  0  ->  (
1  <  ( # `  A
)  ->  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) ) ) )
30 orc 385 . . . . . . . . . . 11  |-  ( (
# `  B )  =  1  ->  (
( # `  B )  =  1  \/  B  =  (/) ) )
3130olcd 393 . . . . . . . . . 10  |-  ( (
# `  B )  =  1  ->  (
( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) )
3231a1d 25 . . . . . . . . 9  |-  ( (
# `  B )  =  1  ->  ( V FriendGrph  E  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) )
3332a1d 25 . . . . . . . 8  |-  ( (
# `  B )  =  1  ->  (
( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( ( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) ) ) )
3433a1d 25 . . . . . . 7  |-  ( (
# `  B )  =  1  ->  (
1  <  ( # `  A
)  ->  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) ) ) )
351, 2frgrawopreglem5 30779 . . . . . . . . . . . 12  |-  ( ( V FriendGrph  E  /\  1  <  ( # `  A
)  /\  1  <  (
# `  B )
)  ->  E. a  e.  A  E. x  e.  A  E. b  e.  B  E. y  e.  B  ( (
b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )
36353expb 1189 . . . . . . . . . . 11  |-  ( ( V FriendGrph  E  /\  (
1  <  ( # `  A
)  /\  1  <  (
# `  B )
) )  ->  E. a  e.  A  E. x  e.  A  E. b  e.  B  E. y  e.  B  ( (
b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )
37 frisusgra 30722 . . . . . . . . . . . . . . . 16  |-  ( V FriendGrph  E  ->  V USGrph  E )
38 simplll 757 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( V USGrph  E  /\  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  /\  (
a  e.  A  /\  x  e.  A )
)  /\  ( b  e.  B  /\  y  e.  B ) )  ->  V USGrph  E )
39 elrabi 3211 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( a  e.  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }  ->  a  e.  V )
4039, 1eleq2s 2559 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( a  e.  A  ->  a  e.  V )
4140ad2antrl 727 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( V USGrph  E  /\  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  /\  (
a  e.  A  /\  x  e.  A )
)  ->  a  e.  V )
4241adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( V USGrph  E  /\  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  /\  (
a  e.  A  /\  x  e.  A )
)  /\  ( b  e.  B  /\  y  e.  B ) )  -> 
a  e.  V )
431rabeq2i 3065 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  e.  A  <->  ( x  e.  V  /\  (
( V VDeg  E ) `  x )  =  K ) )
4443simplbi 460 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  e.  A  ->  x  e.  V )
4544ad2antll 728 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( V USGrph  E  /\  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  /\  (
a  e.  A  /\  x  e.  A )
)  ->  x  e.  V )
4645adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( V USGrph  E  /\  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  /\  (
a  e.  A  /\  x  e.  A )
)  /\  ( b  e.  B  /\  y  e.  B ) )  ->  x  e.  V )
47 simpr1r 1046 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( V USGrph  E  /\  (
( b  =/=  y  /\  a  =/=  x
)  /\  ( {
a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E )  /\  ( { a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  ->  a  =/=  x
)
4847adantr 465 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( V USGrph  E  /\  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  /\  (
a  e.  A  /\  x  e.  A )
)  ->  a  =/=  x )
4948adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( V USGrph  E  /\  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  /\  (
a  e.  A  /\  x  e.  A )
)  /\  ( b  e.  B  /\  y  e.  B ) )  -> 
a  =/=  x )
5042, 46, 493jca 1168 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( V USGrph  E  /\  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  /\  (
a  e.  A  /\  x  e.  A )
)  /\  ( b  e.  B  /\  y  e.  B ) )  -> 
( a  e.  V  /\  x  e.  V  /\  a  =/=  x
) )
512eleq2i 2529 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( b  e.  B  <->  b  e.  ( V  \  A ) )
52 eldif 3436 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( b  e.  ( V  \  A )  <->  ( b  e.  V  /\  -.  b  e.  A ) )
5351, 52bitri 249 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( b  e.  B  <->  ( b  e.  V  /\  -.  b  e.  A ) )
5453simplbi 460 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( b  e.  B  ->  b  e.  V )
5554ad2antrl 727 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( V USGrph  E  /\  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  /\  (
a  e.  A  /\  x  e.  A )
)  /\  ( b  e.  B  /\  y  e.  B ) )  -> 
b  e.  V )
562eleq2i 2529 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  B  <->  y  e.  ( V  \  A ) )
57 eldif 3436 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  ( V  \  A )  <->  ( y  e.  V  /\  -.  y  e.  A ) )
5856, 57bitri 249 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  B  <->  ( y  e.  V  /\  -.  y  e.  A ) )
5958simplbi 460 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  B  ->  y  e.  V )
6059ad2antll 728 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( V USGrph  E  /\  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  /\  (
a  e.  A  /\  x  e.  A )
)  /\  ( b  e.  B  /\  y  e.  B ) )  -> 
y  e.  V )
61 simpr1l 1045 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( V USGrph  E  /\  (
( b  =/=  y  /\  a  =/=  x
)  /\  ( {
a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E )  /\  ( { a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  ->  b  =/=  y
)
6261adantr 465 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( V USGrph  E  /\  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  /\  (
a  e.  A  /\  x  e.  A )
)  ->  b  =/=  y )
6362adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( V USGrph  E  /\  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  /\  (
a  e.  A  /\  x  e.  A )
)  /\  ( b  e.  B  /\  y  e.  B ) )  -> 
b  =/=  y )
6455, 60, 633jca 1168 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( V USGrph  E  /\  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  /\  (
a  e.  A  /\  x  e.  A )
)  /\  ( b  e.  B  /\  y  e.  B ) )  -> 
( b  e.  V  /\  y  e.  V  /\  b  =/=  y
) )
65 prcom 4051 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  { x ,  b }  =  { b ,  x }
6665eleq1i 2528 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( { x ,  b }  e.  ran  E  <->  { b ,  x }  e.  ran  E )
6766biimpi 194 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( { x ,  b }  e.  ran  E  ->  { b ,  x }  e.  ran  E )
6867anim2i 569 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  ->  ( {
a ,  b }  e.  ran  E  /\  { b ,  x }  e.  ran  E ) )
69 prcom 4051 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  { a ,  y }  =  { y ,  a }
7069eleq1i 2528 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( { a ,  y }  e.  ran  E  <->  { y ,  a }  e.  ran  E )
7170biimpi 194 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( { a ,  y }  e.  ran  E  ->  { y ,  a }  e.  ran  E
)
7271anim1i 568 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( { a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E
)  ->  ( {
y ,  a }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) )
7372ancomd 451 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( { a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E
)  ->  ( {
x ,  y }  e.  ran  E  /\  { y ,  a }  e.  ran  E ) )
7468, 73anim12i 566 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E )  /\  ( { a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) )  ->  ( ( { a ,  b }  e.  ran  E  /\  { b ,  x }  e.  ran  E )  /\  ( { x ,  y }  e.  ran  E  /\  { y ,  a }  e.  ran  E ) ) )
75743adant1 1006 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( b  =/=  y  /\  a  =/=  x
)  /\  ( {
a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E )  /\  ( { a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) )  -> 
( ( { a ,  b }  e.  ran  E  /\  { b ,  x }  e.  ran  E )  /\  ( { x ,  y }  e.  ran  E  /\  { y ,  a }  e.  ran  E
) ) )
7675ad3antlr 730 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( V USGrph  E  /\  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  /\  (
a  e.  A  /\  x  e.  A )
)  /\  ( b  e.  B  /\  y  e.  B ) )  -> 
( ( { a ,  b }  e.  ran  E  /\  { b ,  x }  e.  ran  E )  /\  ( { x ,  y }  e.  ran  E  /\  { y ,  a }  e.  ran  E
) ) )
77 4cyclusnfrgra 30749 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( V USGrph  E  /\  (
a  e.  V  /\  x  e.  V  /\  a  =/=  x )  /\  ( b  e.  V  /\  y  e.  V  /\  b  =/=  y
) )  ->  (
( ( { a ,  b }  e.  ran  E  /\  { b ,  x }  e.  ran  E )  /\  ( { x ,  y }  e.  ran  E  /\  { y ,  a }  e.  ran  E
) )  ->  -.  V FriendGrph  E ) )
7877imp 429 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( V USGrph  E  /\  ( a  e.  V  /\  x  e.  V  /\  a  =/=  x
)  /\  ( b  e.  V  /\  y  e.  V  /\  b  =/=  y ) )  /\  ( ( { a ,  b }  e.  ran  E  /\  { b ,  x }  e.  ran  E )  /\  ( { x ,  y }  e.  ran  E  /\  { y ,  a }  e.  ran  E
) ) )  ->  -.  V FriendGrph  E )
7938, 50, 64, 76, 78syl31anc 1222 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( V USGrph  E  /\  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  /\  (
a  e.  A  /\  x  e.  A )
)  /\  ( b  e.  B  /\  y  e.  B ) )  ->  -.  V FriendGrph  E )
8079pm2.21d 106 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( V USGrph  E  /\  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )  /\  (
a  e.  A  /\  x  e.  A )
)  /\  ( b  e.  B  /\  y  e.  B ) )  -> 
( V FriendGrph  E  ->  (
( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) ) )
8180exp41 610 . . . . . . . . . . . . . . . . 17  |-  ( V USGrph  E  ->  ( ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) )  ->  ( (
a  e.  A  /\  x  e.  A )  ->  ( ( b  e.  B  /\  y  e.  B )  ->  ( V FriendGrph  E  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) ) ) ) )
8281com25 91 . . . . . . . . . . . . . . . 16  |-  ( V USGrph  E  ->  ( V FriendGrph  E  -> 
( ( a  e.  A  /\  x  e.  A )  ->  (
( b  e.  B  /\  y  e.  B
)  ->  ( (
( b  =/=  y  /\  a  =/=  x
)  /\  ( {
a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E )  /\  ( { a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) )  -> 
( ( ( # `  A )  =  1  \/  A  =  (/) )  \/  ( ( # `
 B )  =  1  \/  B  =  (/) ) ) ) ) ) ) )
8337, 82mpcom 36 . . . . . . . . . . . . . . 15  |-  ( V FriendGrph  E  ->  ( ( a  e.  A  /\  x  e.  A )  ->  (
( b  e.  B  /\  y  e.  B
)  ->  ( (
( b  =/=  y  /\  a  =/=  x
)  /\  ( {
a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E )  /\  ( { a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) )  -> 
( ( ( # `  A )  =  1  \/  A  =  (/) )  \/  ( ( # `
 B )  =  1  \/  B  =  (/) ) ) ) ) ) )
8483imp 429 . . . . . . . . . . . . . 14  |-  ( ( V FriendGrph  E  /\  (
a  e.  A  /\  x  e.  A )
)  ->  ( (
b  e.  B  /\  y  e.  B )  ->  ( ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) )  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) ) )
8584rexlimdvv 2943 . . . . . . . . . . . . 13  |-  ( ( V FriendGrph  E  /\  (
a  e.  A  /\  x  e.  A )
)  ->  ( E. b  e.  B  E. y  e.  B  (
( b  =/=  y  /\  a  =/=  x
)  /\  ( {
a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E )  /\  ( { a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) )  -> 
( ( ( # `  A )  =  1  \/  A  =  (/) )  \/  ( ( # `
 B )  =  1  \/  B  =  (/) ) ) ) )
8685rexlimdvva 2944 . . . . . . . . . . . 12  |-  ( V FriendGrph  E  ->  ( E. a  e.  A  E. x  e.  A  E. b  e.  B  E. y  e.  B  ( (
b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) )  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) )
8786adantr 465 . . . . . . . . . . 11  |-  ( ( V FriendGrph  E  /\  (
1  <  ( # `  A
)  /\  1  <  (
# `  B )
) )  ->  ( E. a  e.  A  E. x  e.  A  E. b  e.  B  E. y  e.  B  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  /\  ( {
a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) )  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) )
8836, 87mpd 15 . . . . . . . . . 10  |-  ( ( V FriendGrph  E  /\  (
1  <  ( # `  A
)  /\  1  <  (
# `  B )
) )  ->  (
( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) )
8988expcom 435 . . . . . . . . 9  |-  ( ( 1  <  ( # `  A )  /\  1  <  ( # `  B
) )  ->  ( V FriendGrph  E  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) )
9089a1d 25 . . . . . . . 8  |-  ( ( 1  <  ( # `  A )  /\  1  <  ( # `  B
) )  ->  (
( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( ( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) ) ) )
9190expcom 435 . . . . . . 7  |-  ( 1  <  ( # `  B
)  ->  ( 1  <  ( # `  A
)  ->  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) ) ) )
9229, 34, 913jaoi 1282 . . . . . 6  |-  ( ( ( # `  B
)  =  0  \/  ( # `  B
)  =  1  \/  1  <  ( # `  B ) )  -> 
( 1  <  ( # `
 A )  -> 
( ( A  e. 
_V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( ( (
# `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) ) ) )
9392com12 31 . . . . 5  |-  ( 1  <  ( # `  A
)  ->  ( (
( # `  B )  =  0  \/  ( # `
 B )  =  1  \/  1  < 
( # `  B ) )  ->  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) ) ) )
9415, 20, 933jaoi 1282 . . . 4  |-  ( ( ( # `  A
)  =  0  \/  ( # `  A
)  =  1  \/  1  <  ( # `  A ) )  -> 
( ( ( # `  B )  =  0  \/  ( # `  B
)  =  1  \/  1  <  ( # `  B ) )  -> 
( ( A  e. 
_V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( ( (
# `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) ) ) )
9594imp 429 . . 3  |-  ( ( ( ( # `  A
)  =  0  \/  ( # `  A
)  =  1  \/  1  <  ( # `  A ) )  /\  ( ( # `  B
)  =  0  \/  ( # `  B
)  =  1  \/  1  <  ( # `  B ) ) )  ->  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) ) )
966, 95mpcom 36 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( ( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) ) )
973, 96mpcom 36 1  |-  ( V FriendGrph  E  ->  ( ( (
# `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 964    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796   {crab 2799   _Vcvv 3068    \ cdif 3423   (/)c0 3735   {cpr 3977   class class class wbr 4390   ran crn 4939   ` cfv 5516  (class class class)co 6190   0cc0 9383   1c1 9384    < clt 9519   #chash 12204   USGrph cusg 23399   VDeg cvdg 23698   FriendGrph cfrgra 30718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-n0 10681  df-z 10748  df-uz 10963  df-xadd 11191  df-fz 11539  df-hash 12205  df-usgra 23401  df-nbgra 23467  df-vdgr 23699  df-frgra 30719
This theorem is referenced by:  frgraregorufr0  30783
  Copyright terms: Public domain W3C validator