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 28511
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 28506 . 2  |-  ( V FriendGrph  E  ->  ( A  e. 
_V  /\  B  e.  _V ) )
4 hashv01gt1 11634 . . . 4  |-  ( A  e.  _V  ->  (
( # `  A )  =  0  \/  ( # `
 A )  =  1  \/  1  < 
( # `  A ) ) )
5 hashv01gt1 11634 . . . 4  |-  ( B  e.  _V  ->  (
( # `  B )  =  0  \/  ( # `
 B )  =  1  \/  1  < 
( # `  B ) ) )
64, 5anim12i 551 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( ( # `  A )  =  0  \/  ( # `  A
)  =  1  \/  1  <  ( # `  A ) )  /\  ( ( # `  B
)  =  0  \/  ( # `  B
)  =  1  \/  1  <  ( # `  B ) ) ) )
7 hasheq0 11649 . . . . . . . . . . . . 13  |-  ( A  e.  _V  ->  (
( # `  A )  =  0  <->  A  =  (/) ) )
87biimpd 200 . . . . . . . . . . . 12  |-  ( A  e.  _V  ->  (
( # `  A )  =  0  ->  A  =  (/) ) )
98adantr 453 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( # `  A
)  =  0  ->  A  =  (/) ) )
109impcom 421 . . . . . . . . . 10  |-  ( ( ( # `  A
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  A  =  (/) )
1110olcd 384 . . . . . . . . 9  |-  ( ( ( # `  A
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( ( # `
 A )  =  1  \/  A  =  (/) ) )
1211orcd 383 . . . . . . . 8  |-  ( ( ( # `  A
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) )
1312a1d 24 . . . . . . 7  |-  ( ( ( # `  A
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( V FriendGrph  E  ->  ( ( (
# `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) )
1413ex 425 . . . . . 6  |-  ( (
# `  A )  =  0  ->  (
( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( ( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) ) ) )
1514a1d 24 . . . . 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 376 . . . . . . . . 9  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  =  1  \/  A  =  (/) ) )
1716orcd 383 . . . . . . . 8  |-  ( (
# `  A )  =  1  ->  (
( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) )
1817a1d 24 . . . . . . 7  |-  ( (
# `  A )  =  1  ->  ( V FriendGrph  E  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) )
1918a1d 24 . . . . . 6  |-  ( (
# `  A )  =  1  ->  (
( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( ( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) ) ) )
2019a1d 24 . . . . 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 11649 . . . . . . . . . . . . . . 15  |-  ( B  e.  _V  ->  (
( # `  B )  =  0  <->  B  =  (/) ) )
2221biimpd 200 . . . . . . . . . . . . . 14  |-  ( B  e.  _V  ->  (
( # `  B )  =  0  ->  B  =  (/) ) )
2322adantl 454 . . . . . . . . . . . . 13  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( # `  B
)  =  0  ->  B  =  (/) ) )
2423impcom 421 . . . . . . . . . . . 12  |-  ( ( ( # `  B
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  B  =  (/) )
2524olcd 384 . . . . . . . . . . 11  |-  ( ( ( # `  B
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( ( # `
 B )  =  1  \/  B  =  (/) ) )
2625olcd 384 . . . . . . . . . 10  |-  ( ( ( # `  B
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) )
2726a1d 24 . . . . . . . . 9  |-  ( ( ( # `  B
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( V FriendGrph  E  ->  ( ( (
# `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) )
2827ex 425 . . . . . . . 8  |-  ( (
# `  B )  =  0  ->  (
( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( ( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) ) ) )
2928a1d 24 . . . . . . 7  |-  ( (
# `  B )  =  0  ->  (
1  <  ( # `  A
)  ->  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) ) ) )
30 orc 376 . . . . . . . . . . 11  |-  ( (
# `  B )  =  1  ->  (
( # `  B )  =  1  \/  B  =  (/) ) )
3130olcd 384 . . . . . . . . . 10  |-  ( (
# `  B )  =  1  ->  (
( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) )
3231a1d 24 . . . . . . . . 9  |-  ( (
# `  B )  =  1  ->  ( V FriendGrph  E  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) )
3332a1d 24 . . . . . . . 8  |-  ( (
# `  B )  =  1  ->  (
( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( ( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) ) ) )
3433a1d 24 . . . . . . 7  |-  ( (
# `  B )  =  1  ->  (
1  <  ( # `  A
)  ->  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) ) ) )
351, 2frgrawopreglem5 28510 . . . . . . . . . . . 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 1155 . . . . . . . . . . 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 28455 . . . . . . . . . . . . . . . 16  |-  ( V FriendGrph  E  ->  V USGrph  E )
38 simplll 736 . . . . . . . . . . . . . . . . . . . 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 3092 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( a  e.  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }  ->  a  e.  V )
4039, 1eleq2s 2530 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( a  e.  A  ->  a  e.  V )
4140ad2antrl 710 . . . . . . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . . . . . . 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 2955 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  e.  A  <->  ( x  e.  V  /\  (
( V VDeg  E ) `  x )  =  K ) )
4443simplbi 448 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  e.  A  ->  x  e.  V )
4544ad2antll 711 . . . . . . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . . . . . . 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 1016 . . . . . . . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . . . . . . 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 2502 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( b  e.  B  <->  b  e.  ( V  \  A ) )
52 eldif 3332 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( b  e.  ( V  \  A )  <->  ( b  e.  V  /\  -.  b  e.  A ) )
5351, 52bitri 242 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( b  e.  B  <->  ( b  e.  V  /\  -.  b  e.  A ) )
5453simplbi 448 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( b  e.  B  ->  b  e.  V )
5554ad2antrl 710 . . . . . . . . . . . . . . . . . . . . 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 2502 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  B  <->  y  e.  ( V  \  A ) )
57 eldif 3332 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  ( V  \  A )  <->  ( y  e.  V  /\  -.  y  e.  A ) )
5856, 57bitri 242 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  B  <->  ( y  e.  V  /\  -.  y  e.  A ) )
5958simplbi 448 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  B  ->  y  e.  V )
6059ad2antll 711 . . . . . . . . . . . . . . . . . . . . 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 1015 . . . . . . . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . . . . . . 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 3884 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  { x ,  b }  =  { b ,  x }
6665eleq1i 2501 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( { x ,  b }  e.  ran  E  <->  { b ,  x }  e.  ran  E )
6766biimpi 188 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( { x ,  b }  e.  ran  E  ->  { b ,  x }  e.  ran  E )
6867anim2i 554 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  ->  ( {
a ,  b }  e.  ran  E  /\  { b ,  x }  e.  ran  E ) )
69 prcom 3884 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  { a ,  y }  =  { y ,  a }
7069eleq1i 2501 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( { a ,  y }  e.  ran  E  <->  { y ,  a }  e.  ran  E )
7170biimpi 188 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( { a ,  y }  e.  ran  E  ->  { y ,  a }  e.  ran  E
)
7271anim1i 553 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( { a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E
)  ->  ( {
y ,  a }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) )
7372ancomd 440 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( { a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E
)  ->  ( {
x ,  y }  e.  ran  E  /\  { y ,  a }  e.  ran  E ) )
7468, 73anim12i 551 . . . . . . . . . . . . . . . . . . . . . 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 976 . . . . . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . . . . . . 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 28482 . . . . . . . . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . . . . . . . . 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 1188 . . . . . . . . . . . . . . . . . . 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 101 . . . . . . . . . . . . . . . . . 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 595 . . . . . . . . . . . . . . . . 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 88 . . . . . . . . . . . . . . . 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 35 . . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . . 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 2838 . . . . . . . . . . . . 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 2839 . . . . . . . . . . . 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 453 . . . . . . . . . . 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 426 . . . . . . . . 9  |-  ( ( 1  <  ( # `  A )  /\  1  <  ( # `  B
) )  ->  ( V FriendGrph  E  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) )
9089a1d 24 . . . . . . . 8  |-  ( ( 1  <  ( # `  A )  /\  1  <  ( # `  B
) )  ->  (
( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( ( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) ) ) )
9190expcom 426 . . . . . . 7  |-  ( 1  <  ( # `  B
)  ->  ( 1  <  ( # `  A
)  ->  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) ) ) )
9229, 34, 913jaoi 1248 . . . . . 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 30 . . . . 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 1248 . . . 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 420 . . 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 35 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( ( ( # `  A
)  =  1  \/  A  =  (/) )  \/  ( ( # `  B
)  =  1  \/  B  =  (/) ) ) ) )
973, 96mpcom 35 1  |-  ( V FriendGrph  E  ->  ( ( (
# `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 359    /\ wa 360    \/ w3o 936    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   {crab 2711   _Vcvv 2958    \ cdif 3319   (/)c0 3630   {cpr 3817   class class class wbr 4215   ran crn 4882   ` cfv 5457  (class class class)co 6084   0cc0 8995   1c1 8996    < clt 9125   #chash 11623   USGrph cusg 21370   VDeg cvdg 21669   FriendGrph cfrgra 28451
This theorem is referenced by:  frgraregorufr0  28514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-xadd 10716  df-fz 11049  df-hash 11624  df-usgra 21372  df-nbgra 21438  df-vdgr 21670  df-frgra 28452
  Copyright terms: Public domain W3C validator