MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankcf Structured version   Visualization version   Unicode version

Theorem rankcf 9233
Description: Any set must be at least as large as the cofinality of its rank, because the ranks of the elements of 
A form a cofinal map into  ( rank `  A
). (Contributed by Mario Carneiro, 27-May-2013.)
Assertion
Ref Expression
rankcf  |-  -.  A  ~<  ( cf `  ( rank `  A ) )

Proof of Theorem rankcf
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rankon 8297 . . 3  |-  ( rank `  A )  e.  On
2 onzsl 6705 . . 3  |-  ( (
rank `  A )  e.  On  <->  ( ( rank `  A )  =  (/)  \/ 
E. x  e.  On  ( rank `  A )  =  suc  x  \/  (
( rank `  A )  e.  _V  /\  Lim  ( rank `  A ) ) ) )
31, 2mpbi 213 . 2  |-  ( (
rank `  A )  =  (/)  \/  E. x  e.  On  ( rank `  A
)  =  suc  x  \/  ( ( rank `  A
)  e.  _V  /\  Lim  ( rank `  A
) ) )
4 sdom0 7735 . . . 4  |-  -.  A  ~< 
(/)
5 fveq2 5892 . . . . . 6  |-  ( (
rank `  A )  =  (/)  ->  ( cf `  ( rank `  A
) )  =  ( cf `  (/) ) )
6 cf0 8712 . . . . . 6  |-  ( cf `  (/) )  =  (/)
75, 6syl6eq 2512 . . . . 5  |-  ( (
rank `  A )  =  (/)  ->  ( cf `  ( rank `  A
) )  =  (/) )
87breq2d 4430 . . . 4  |-  ( (
rank `  A )  =  (/)  ->  ( A  ~<  ( cf `  ( rank `  A ) )  <-> 
A  ~<  (/) ) )
94, 8mtbiri 309 . . 3  |-  ( (
rank `  A )  =  (/)  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
10 fveq2 5892 . . . . . . 7  |-  ( (
rank `  A )  =  suc  x  ->  ( cf `  ( rank `  A
) )  =  ( cf `  suc  x
) )
11 cfsuc 8718 . . . . . . 7  |-  ( x  e.  On  ->  ( cf `  suc  x )  =  1o )
1210, 11sylan9eqr 2518 . . . . . 6  |-  ( ( x  e.  On  /\  ( rank `  A )  =  suc  x )  -> 
( cf `  ( rank `  A ) )  =  1o )
13 nsuceq0 5526 . . . . . . . . 9  |-  suc  x  =/=  (/)
14 neeq1 2698 . . . . . . . . 9  |-  ( (
rank `  A )  =  suc  x  ->  (
( rank `  A )  =/=  (/)  <->  suc  x  =/=  (/) ) )
1513, 14mpbiri 241 . . . . . . . 8  |-  ( (
rank `  A )  =  suc  x  ->  ( rank `  A )  =/=  (/) )
16 fveq2 5892 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  ( rank `  A )  =  (
rank `  (/) ) )
17 0elon 5499 . . . . . . . . . . . . 13  |-  (/)  e.  On
18 r1fnon 8269 . . . . . . . . . . . . . 14  |-  R1  Fn  On
19 fndm 5701 . . . . . . . . . . . . . 14  |-  ( R1  Fn  On  ->  dom  R1  =  On )
2018, 19ax-mp 5 . . . . . . . . . . . . 13  |-  dom  R1  =  On
2117, 20eleqtrri 2539 . . . . . . . . . . . 12  |-  (/)  e.  dom  R1
22 rankonid 8331 . . . . . . . . . . . 12  |-  ( (/)  e.  dom  R1  <->  ( rank `  (/) )  =  (/) )
2321, 22mpbi 213 . . . . . . . . . . 11  |-  ( rank `  (/) )  =  (/)
2416, 23syl6eq 2512 . . . . . . . . . 10  |-  ( A  =  (/)  ->  ( rank `  A )  =  (/) )
2524necon3i 2668 . . . . . . . . 9  |-  ( (
rank `  A )  =/=  (/)  ->  A  =/=  (/) )
26 rankvaln 8301 . . . . . . . . . . 11  |-  ( -.  A  e.  U. ( R1 " On )  -> 
( rank `  A )  =  (/) )
2726necon1ai 2663 . . . . . . . . . 10  |-  ( (
rank `  A )  =/=  (/)  ->  A  e.  U. ( R1 " On ) )
28 breq2 4422 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( 1o 
~<_  y  <->  1o  ~<_  A )
)
29 neeq1 2698 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
y  =/=  (/)  <->  A  =/=  (/) ) )
30 0sdom1dom 7801 . . . . . . . . . . . 12  |-  ( (/)  ~< 
y  <->  1o  ~<_  y )
31 vex 3060 . . . . . . . . . . . . 13  |-  y  e. 
_V
32310sdom 7734 . . . . . . . . . . . 12  |-  ( (/)  ~< 
y  <->  y  =/=  (/) )
3330, 32bitr3i 259 . . . . . . . . . . 11  |-  ( 1o  ~<_  y  <->  y  =/=  (/) )
3428, 29, 33vtoclbg 3120 . . . . . . . . . 10  |-  ( A  e.  U. ( R1
" On )  -> 
( 1o  ~<_  A  <->  A  =/=  (/) ) )
3527, 34syl 17 . . . . . . . . 9  |-  ( (
rank `  A )  =/=  (/)  ->  ( 1o  ~<_  A 
<->  A  =/=  (/) ) )
3625, 35mpbird 240 . . . . . . . 8  |-  ( (
rank `  A )  =/=  (/)  ->  1o  ~<_  A )
3715, 36syl 17 . . . . . . 7  |-  ( (
rank `  A )  =  suc  x  ->  1o  ~<_  A )
3837adantl 472 . . . . . 6  |-  ( ( x  e.  On  /\  ( rank `  A )  =  suc  x )  ->  1o 
~<_  A )
3912, 38eqbrtrd 4439 . . . . 5  |-  ( ( x  e.  On  /\  ( rank `  A )  =  suc  x )  -> 
( cf `  ( rank `  A ) )  ~<_  A )
4039rexlimiva 2887 . . . 4  |-  ( E. x  e.  On  ( rank `  A )  =  suc  x  ->  ( cf `  ( rank `  A
) )  ~<_  A )
41 domnsym 7729 . . . 4  |-  ( ( cf `  ( rank `  A ) )  ~<_  A  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
4240, 41syl 17 . . 3  |-  ( E. x  e.  On  ( rank `  A )  =  suc  x  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
43 nlim0 5504 . . . . . . . . . . . . . . . . 17  |-  -.  Lim  (/)
44 limeq 5458 . . . . . . . . . . . . . . . . 17  |-  ( (
rank `  A )  =  (/)  ->  ( Lim  ( rank `  A )  <->  Lim  (/) ) )
4543, 44mtbiri 309 . . . . . . . . . . . . . . . 16  |-  ( (
rank `  A )  =  (/)  ->  -.  Lim  ( rank `  A ) )
4626, 45syl 17 . . . . . . . . . . . . . . 15  |-  ( -.  A  e.  U. ( R1 " On )  ->  -.  Lim  ( rank `  A
) )
4746con4i 135 . . . . . . . . . . . . . 14  |-  ( Lim  ( rank `  A
)  ->  A  e.  U. ( R1 " On ) )
48 r1elssi 8307 . . . . . . . . . . . . . 14  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  U. ( R1
" On ) )
4947, 48syl 17 . . . . . . . . . . . . 13  |-  ( Lim  ( rank `  A
)  ->  A  C_  U. ( R1 " On ) )
5049sselda 3444 . . . . . . . . . . . 12  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  x  e.  U. ( R1 " On ) )
51 ranksnb 8329 . . . . . . . . . . . 12  |-  ( x  e.  U. ( R1
" On )  -> 
( rank `  { x } )  =  suc  ( rank `  x )
)
5250, 51syl 17 . . . . . . . . . . 11  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  ( rank `  { x }
)  =  suc  ( rank `  x ) )
53 rankelb 8326 . . . . . . . . . . . . . 14  |-  ( A  e.  U. ( R1
" On )  -> 
( x  e.  A  ->  ( rank `  x
)  e.  ( rank `  A ) ) )
5447, 53syl 17 . . . . . . . . . . . . 13  |-  ( Lim  ( rank `  A
)  ->  ( x  e.  A  ->  ( rank `  x )  e.  (
rank `  A )
) )
55 limsuc 6708 . . . . . . . . . . . . 13  |-  ( Lim  ( rank `  A
)  ->  ( ( rank `  x )  e.  ( rank `  A
)  <->  suc  ( rank `  x
)  e.  ( rank `  A ) ) )
5654, 55sylibd 222 . . . . . . . . . . . 12  |-  ( Lim  ( rank `  A
)  ->  ( x  e.  A  ->  suc  ( rank `  x )  e.  ( rank `  A
) ) )
5756imp 435 . . . . . . . . . . 11  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  suc  ( rank `  x )  e.  ( rank `  A
) )
5852, 57eqeltrd 2540 . . . . . . . . . 10  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  ( rank `  { x }
)  e.  ( rank `  A ) )
59 eleq1a 2535 . . . . . . . . . 10  |-  ( (
rank `  { x } )  e.  (
rank `  A )  ->  ( w  =  (
rank `  { x } )  ->  w  e.  ( rank `  A
) ) )
6058, 59syl 17 . . . . . . . . 9  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  (
w  =  ( rank `  { x } )  ->  w  e.  (
rank `  A )
) )
6160rexlimdva 2891 . . . . . . . 8  |-  ( Lim  ( rank `  A
)  ->  ( E. x  e.  A  w  =  ( rank `  {
x } )  ->  w  e.  ( rank `  A ) ) )
6261abssdv 3515 . . . . . . 7  |-  ( Lim  ( rank `  A
)  ->  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  C_  ( rank `  A ) )
63 snex 4658 . . . . . . . . . . . . 13  |-  { x }  e.  _V
6463dfiun2 4326 . . . . . . . . . . . 12  |-  U_ x  e.  A  { x }  =  U. { y  |  E. x  e.  A  y  =  {
x } }
65 iunid 4347 . . . . . . . . . . . 12  |-  U_ x  e.  A  { x }  =  A
6664, 65eqtr3i 2486 . . . . . . . . . . 11  |-  U. {
y  |  E. x  e.  A  y  =  { x } }  =  A
6766fveq2i 5895 . . . . . . . . . 10  |-  ( rank `  U. { y  |  E. x  e.  A  y  =  { x } } )  =  (
rank `  A )
6848sselda 3444 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  U. ( R1 " On )  /\  x  e.  A )  ->  x  e.  U. ( R1 " On ) )
69 snwf 8311 . . . . . . . . . . . . . . 15  |-  ( x  e.  U. ( R1
" On )  ->  { x }  e.  U. ( R1 " On ) )
70 eleq1a 2535 . . . . . . . . . . . . . . 15  |-  ( { x }  e.  U. ( R1 " On )  ->  ( y  =  { x }  ->  y  e.  U. ( R1
" On ) ) )
7168, 69, 703syl 18 . . . . . . . . . . . . . 14  |-  ( ( A  e.  U. ( R1 " On )  /\  x  e.  A )  ->  ( y  =  {
x }  ->  y  e.  U. ( R1 " On ) ) )
7271rexlimdva 2891 . . . . . . . . . . . . 13  |-  ( A  e.  U. ( R1
" On )  -> 
( E. x  e.  A  y  =  {
x }  ->  y  e.  U. ( R1 " On ) ) )
7372abssdv 3515 . . . . . . . . . . . 12  |-  ( A  e.  U. ( R1
" On )  ->  { y  |  E. x  e.  A  y  =  { x } }  C_ 
U. ( R1 " On ) )
74 abrexexg 6800 . . . . . . . . . . . . 13  |-  ( A  e.  U. ( R1
" On )  ->  { y  |  E. x  e.  A  y  =  { x } }  e.  _V )
75 eleq1 2528 . . . . . . . . . . . . . 14  |-  ( z  =  { y  |  E. x  e.  A  y  =  { x } }  ->  ( z  e.  U. ( R1
" On )  <->  { y  |  E. x  e.  A  y  =  { x } }  e.  U. ( R1 " On ) ) )
76 sseq1 3465 . . . . . . . . . . . . . 14  |-  ( z  =  { y  |  E. x  e.  A  y  =  { x } }  ->  ( z 
C_  U. ( R1 " On )  <->  { y  |  E. x  e.  A  y  =  { x } }  C_ 
U. ( R1 " On ) ) )
77 vex 3060 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
7877r1elss 8308 . . . . . . . . . . . . . 14  |-  ( z  e.  U. ( R1
" On )  <->  z  C_  U. ( R1 " On ) )
7975, 76, 78vtoclbg 3120 . . . . . . . . . . . . 13  |-  ( { y  |  E. x  e.  A  y  =  { x } }  e.  _V  ->  ( {
y  |  E. x  e.  A  y  =  { x } }  e.  U. ( R1 " On )  <->  { y  |  E. x  e.  A  y  =  { x } }  C_ 
U. ( R1 " On ) ) )
8074, 79syl 17 . . . . . . . . . . . 12  |-  ( A  e.  U. ( R1
" On )  -> 
( { y  |  E. x  e.  A  y  =  { x } }  e.  U. ( R1 " On )  <->  { y  |  E. x  e.  A  y  =  { x } }  C_  U. ( R1 " On ) ) )
8173, 80mpbird 240 . . . . . . . . . . 11  |-  ( A  e.  U. ( R1
" On )  ->  { y  |  E. x  e.  A  y  =  { x } }  e.  U. ( R1 " On ) )
82 rankuni2b 8355 . . . . . . . . . . 11  |-  ( { y  |  E. x  e.  A  y  =  { x } }  e.  U. ( R1 " On )  ->  ( rank `  U. { y  |  E. x  e.  A  y  =  { x } } )  =  U_ z  e.  { y  |  E. x  e.  A  y  =  { x } }  ( rank `  z ) )
8381, 82syl 17 . . . . . . . . . 10  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  U. { y  |  E. x  e.  A  y  =  {
x } } )  =  U_ z  e. 
{ y  |  E. x  e.  A  y  =  { x } } 
( rank `  z )
)
8467, 83syl5eqr 2510 . . . . . . . . 9  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  U_ z  e.  {
y  |  E. x  e.  A  y  =  { x } } 
( rank `  z )
)
85 fvex 5902 . . . . . . . . . . 11  |-  ( rank `  z )  e.  _V
8685dfiun2 4326 . . . . . . . . . 10  |-  U_ z  e.  { y  |  E. x  e.  A  y  =  { x } } 
( rank `  z )  =  U. { w  |  E. z  e.  {
y  |  E. x  e.  A  y  =  { x } }
w  =  ( rank `  z ) }
87 fveq2 5892 . . . . . . . . . . . 12  |-  ( z  =  { x }  ->  ( rank `  z
)  =  ( rank `  { x } ) )
8863, 87abrexco 6179 . . . . . . . . . . 11  |-  { w  |  E. z  e.  {
y  |  E. x  e.  A  y  =  { x } }
w  =  ( rank `  z ) }  =  { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }
8988unieqi 4221 . . . . . . . . . 10  |-  U. {
w  |  E. z  e.  { y  |  E. x  e.  A  y  =  { x } }
w  =  ( rank `  z ) }  =  U. { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }
9086, 89eqtri 2484 . . . . . . . . 9  |-  U_ z  e.  { y  |  E. x  e.  A  y  =  { x } } 
( rank `  z )  =  U. { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }
9184, 90syl6req 2513 . . . . . . . 8  |-  ( A  e.  U. ( R1
" On )  ->  U. { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }  =  ( rank `  A
) )
9247, 91syl 17 . . . . . . 7  |-  ( Lim  ( rank `  A
)  ->  U. { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  =  ( rank `  A ) )
93 fvex 5902 . . . . . . . 8  |-  ( rank `  A )  e.  _V
9493cfslb 8727 . . . . . . 7  |-  ( ( Lim  ( rank `  A
)  /\  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  C_  ( rank `  A )  /\  U. { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }  =  ( rank `  A
) )  ->  ( cf `  ( rank `  A
) )  ~<_  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) } )
9562, 92, 94mpd3an23 1375 . . . . . 6  |-  ( Lim  ( rank `  A
)  ->  ( cf `  ( rank `  A
) )  ~<_  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) } )
96 fveq2 5892 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( rank `  y )  =  ( rank `  A
) )
9796fveq2d 5896 . . . . . . . . . 10  |-  ( y  =  A  ->  ( cf `  ( rank `  y
) )  =  ( cf `  ( rank `  A ) ) )
98 breq12 4423 . . . . . . . . . 10  |-  ( ( y  =  A  /\  ( cf `  ( rank `  y ) )  =  ( cf `  ( rank `  A ) ) )  ->  ( y  ~<  ( cf `  ( rank `  y ) )  <-> 
A  ~<  ( cf `  ( rank `  A ) ) ) )
9997, 98mpdan 679 . . . . . . . . 9  |-  ( y  =  A  ->  (
y  ~<  ( cf `  ( rank `  y ) )  <-> 
A  ~<  ( cf `  ( rank `  A ) ) ) )
100 rexeq 3000 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( E. x  e.  y  w  =  ( rank `  { x } )  <->  E. x  e.  A  w  =  ( rank `  { x } ) ) )
101100abbidv 2580 . . . . . . . . . 10  |-  ( y  =  A  ->  { w  |  E. x  e.  y  w  =  ( rank `  { x } ) }  =  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) } )
102 breq12 4423 . . . . . . . . . 10  |-  ( ( { w  |  E. x  e.  y  w  =  ( rank `  {
x } ) }  =  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  /\  y  =  A )  ->  ( { w  |  E. x  e.  y  w  =  ( rank `  {
x } ) }  ~<_  y  <->  { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }  ~<_  A ) )
103101, 102mpancom 680 . . . . . . . . 9  |-  ( y  =  A  ->  ( { w  |  E. x  e.  y  w  =  ( rank `  {
x } ) }  ~<_  y  <->  { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }  ~<_  A ) )
10499, 103imbi12d 326 . . . . . . . 8  |-  ( y  =  A  ->  (
( y  ~<  ( cf `  ( rank `  y
) )  ->  { w  |  E. x  e.  y  w  =  ( rank `  { x } ) }  ~<_  y )  <->  ( A  ~<  ( cf `  ( rank `  A ) )  ->  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  ~<_  A ) ) )
105 eqid 2462 . . . . . . . . . 10  |-  ( x  e.  y  |->  ( rank `  { x } ) )  =  ( x  e.  y  |->  ( rank `  { x } ) )
106105rnmpt 5102 . . . . . . . . 9  |-  ran  (
x  e.  y  |->  (
rank `  { x } ) )  =  { w  |  E. x  e.  y  w  =  ( rank `  {
x } ) }
107 cfon 8716 . . . . . . . . . . 11  |-  ( cf `  ( rank `  y
) )  e.  On
108 sdomdom 7628 . . . . . . . . . . 11  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  y  ~<_  ( cf `  ( rank `  y
) ) )
109 ondomen 8499 . . . . . . . . . . 11  |-  ( ( ( cf `  ( rank `  y ) )  e.  On  /\  y  ~<_  ( cf `  ( rank `  y ) ) )  ->  y  e.  dom  card )
110107, 108, 109sylancr 674 . . . . . . . . . 10  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  y  e.  dom  card )
111 fvex 5902 . . . . . . . . . . . 12  |-  ( rank `  { x } )  e.  _V
112111, 105fnmpti 5732 . . . . . . . . . . 11  |-  ( x  e.  y  |->  ( rank `  { x } ) )  Fn  y
113 dffn4 5826 . . . . . . . . . . 11  |-  ( ( x  e.  y  |->  (
rank `  { x } ) )  Fn  y  <->  ( x  e.  y  |->  ( rank `  {
x } ) ) : y -onto-> ran  (
x  e.  y  |->  (
rank `  { x } ) ) )
114112, 113mpbi 213 . . . . . . . . . 10  |-  ( x  e.  y  |->  ( rank `  { x } ) ) : y -onto-> ran  ( x  e.  y 
|->  ( rank `  {
x } ) )
115 fodomnum 8519 . . . . . . . . . 10  |-  ( y  e.  dom  card  ->  ( ( x  e.  y 
|->  ( rank `  {
x } ) ) : y -onto-> ran  (
x  e.  y  |->  (
rank `  { x } ) )  ->  ran  ( x  e.  y 
|->  ( rank `  {
x } ) )  ~<_  y ) )
116110, 114, 115mpisyl 21 . . . . . . . . 9  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  ran  ( x  e.  y  |->  ( rank `  { x } ) )  ~<_  y )
117106, 116syl5eqbrr 4453 . . . . . . . 8  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  { w  |  E. x  e.  y  w  =  ( rank `  { x } ) }  ~<_  y )
118104, 117vtoclg 3119 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  -> 
( A  ~<  ( cf `  ( rank `  A
) )  ->  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  ~<_  A ) )
11947, 118syl 17 . . . . . 6  |-  ( Lim  ( rank `  A
)  ->  ( A  ~<  ( cf `  ( rank `  A ) )  ->  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  ~<_  A ) )
120 domtr 7653 . . . . . . 7  |-  ( ( ( cf `  ( rank `  A ) )  ~<_  { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }  /\  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  ~<_  A )  -> 
( cf `  ( rank `  A ) )  ~<_  A )
121120, 41syl 17 . . . . . 6  |-  ( ( ( cf `  ( rank `  A ) )  ~<_  { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }  /\  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  ~<_  A )  ->  -.  A  ~<  ( cf `  ( rank `  A
) ) )
12295, 119, 121syl6an 552 . . . . 5  |-  ( Lim  ( rank `  A
)  ->  ( A  ~<  ( cf `  ( rank `  A ) )  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) ) )
123122pm2.01d 174 . . . 4  |-  ( Lim  ( rank `  A
)  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
124123adantl 472 . . 3  |-  ( ( ( rank `  A
)  e.  _V  /\  Lim  ( rank `  A
) )  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
1259, 42, 1243jaoi 1340 . 2  |-  ( ( ( rank `  A
)  =  (/)  \/  E. x  e.  On  ( rank `  A )  =  suc  x  \/  (
( rank `  A )  e.  _V  /\  Lim  ( rank `  A ) ) )  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
1263, 125ax-mp 5 1  |-  -.  A  ~<  ( cf `  ( rank `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    \/ w3o 990    = wceq 1455    e. wcel 1898   {cab 2448    =/= wne 2633   E.wrex 2750   _Vcvv 3057    C_ wss 3416   (/)c0 3743   {csn 3980   U.cuni 4212   U_ciun 4292   class class class wbr 4418    |-> cmpt 4477   dom cdm 4856   ran crn 4857   "cima 4859   Oncon0 5446   Lim wlim 5447   suc csuc 5448    Fn wfn 5600   -onto->wfo 5603   ` cfv 5605   1oc1o 7206    ~<_ cdom 7598    ~< csdm 7599   R1cr1 8264   rankcrnk 8265   cardccrd 8400   cfccf 8402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-se 4816  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-isom 5614  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-er 7394  df-map 7505  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-r1 8266  df-rank 8267  df-card 8404  df-cf 8406  df-acn 8407
This theorem is referenced by:  inatsk  9234  grur1  9276
  Copyright terms: Public domain W3C validator