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Theorem frgrawopreg 25251
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 25246 . 2  |-  ( V FriendGrph  E  ->  ( A  e. 
_V  /\  B  e.  _V ) )
4 hashv01gt1 12400 . . . 4  |-  ( A  e.  _V  ->  (
( # `  A )  =  0  \/  ( # `
 A )  =  1  \/  1  < 
( # `  A ) ) )
5 hashv01gt1 12400 . . . 4  |-  ( B  e.  _V  ->  (
( # `  B )  =  0  \/  ( # `
 B )  =  1  \/  1  < 
( # `  B ) ) )
64, 5anim12i 564 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( ( # `  A )  =  0  \/  ( # `  A
)  =  1  \/  1  <  ( # `  A ) )  /\  ( ( # `  B
)  =  0  \/  ( # `  B
)  =  1  \/  1  <  ( # `  B ) ) ) )
7 hasheq0 12416 . . . . . . . . . . . . 13  |-  ( A  e.  _V  ->  (
( # `  A )  =  0  <->  A  =  (/) ) )
87biimpd 207 . . . . . . . . . . . 12  |-  ( A  e.  _V  ->  (
( # `  A )  =  0  ->  A  =  (/) ) )
98adantr 463 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( # `  A
)  =  0  ->  A  =  (/) ) )
109impcom 428 . . . . . . . . . 10  |-  ( ( ( # `  A
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  A  =  (/) )
1110olcd 391 . . . . . . . . 9  |-  ( ( ( # `  A
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( ( # `
 A )  =  1  \/  A  =  (/) ) )
1211orcd 390 . . . . . . . 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 432 . . . . . 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 383 . . . . . . . . 9  |-  ( (
# `  A )  =  1  ->  (
( # `  A )  =  1  \/  A  =  (/) ) )
1716orcd 390 . . . . . . . 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 12416 . . . . . . . . . . . . . . 15  |-  ( B  e.  _V  ->  (
( # `  B )  =  0  <->  B  =  (/) ) )
2221biimpd 207 . . . . . . . . . . . . . 14  |-  ( B  e.  _V  ->  (
( # `  B )  =  0  ->  B  =  (/) ) )
2322adantl 464 . . . . . . . . . . . . 13  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( # `  B
)  =  0  ->  B  =  (/) ) )
2423impcom 428 . . . . . . . . . . . 12  |-  ( ( ( # `  B
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  B  =  (/) )
2524olcd 391 . . . . . . . . . . 11  |-  ( ( ( # `  B
)  =  0  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( ( # `
 B )  =  1  \/  B  =  (/) ) )
2625olcd 391 . . . . . . . . . 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 432 . . . . . . . 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 383 . . . . . . . . . . 11  |-  ( (
# `  B )  =  1  ->  (
( # `  B )  =  1  \/  B  =  (/) ) )
3130olcd 391 . . . . . . . . . 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 25250 . . . . . . . . . . . 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 1195 . . . . . . . . . . 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 25194 . . . . . . . . . . . . . . . 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 3251 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( a  e.  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }  ->  a  e.  V )
4039, 1eleq2s 2562 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( a  e.  A  ->  a  e.  V )
4140ad2antrl 725 . . . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . . . 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 3103 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  e.  A  <->  ( x  e.  V  /\  (
( V VDeg  E ) `  x )  =  K ) )
4443simplbi 458 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  e.  A  ->  x  e.  V )
4544ad2antll 726 . . . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . . . 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 1052 . . . . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . . . 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 1174 . . . . . . . . . . . . . . . . . . . 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 2532 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( b  e.  B  <->  b  e.  ( V  \  A ) )
52 eldif 3471 . . . . . . . . . . . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( b  e.  B  ->  b  e.  V )
5554ad2antrl 725 . . . . . . . . . . . . . . . . . . . . 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 2532 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  B  <->  y  e.  ( V  \  A ) )
57 eldif 3471 . . . . . . . . . . . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  B  ->  y  e.  V )
6059ad2antll 726 . . . . . . . . . . . . . . . . . . . . 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 1051 . . . . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . . . 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 1174 . . . . . . . . . . . . . . . . . . . 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 4094 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  { x ,  b }  =  { b ,  x }
6665eleq1i 2531 . . . . . . . . . . . . . . . . . . . . . . . . 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 567 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E
)  ->  ( {
a ,  b }  e.  ran  E  /\  { b ,  x }  e.  ran  E ) )
69 prcom 4094 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  { a ,  y }  =  { y ,  a }
7069eleq1i 2531 . . . . . . . . . . . . . . . . . . . . . . . . . 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 566 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( { a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E
)  ->  ( {
y ,  a }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) )
7372ancomd 449 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( { a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E
)  ->  ( {
x ,  y }  e.  ran  E  /\  { y ,  a }  e.  ran  E ) )
7468, 73anim12i 564 . . . . . . . . . . . . . . . . . . . . . 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 1012 . . . . . . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . . . . . 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 25221 . . . . . . . . . . . . . . . . . . . . 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 427 . . . . . . . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . . . . . . . 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 608 . . . . . . . . . . . . . . . . 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 427 . . . . . . . . . . . . . 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 2952 . . . . . . . . . . . . 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 2953 . . . . . . . . . . . 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 463 . . . . . . . . . . 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 433 . . . . . . . . 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 433 . . . . . . 7  |-  ( 1  <  ( # `  B
)  ->  ( 1  <  ( # `  A
)  ->  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( V FriendGrph  E  ->  ( (
( # `  A )  =  1  \/  A  =  (/) )  \/  (
( # `  B )  =  1  \/  B  =  (/) ) ) ) ) ) )
9229, 34, 913jaoi 1289 . . . . . 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 1289 . . . 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 427 . . 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 366    /\ wa 367    \/ w3o 970    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805   {crab 2808   _Vcvv 3106    \ cdif 3458   (/)c0 3783   {cpr 4018   class class class wbr 4439   ran crn 4989   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482    < clt 9617   #chash 12387   USGrph cusg 24532   VDeg cvdg 25095   FriendGrph cfrgra 25190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-xadd 11322  df-fz 11676  df-hash 12388  df-usgra 24535  df-nbgra 24622  df-vdgr 25096  df-frgra 25191
This theorem is referenced by:  frgraregorufr0  25254
  Copyright terms: Public domain W3C validator