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

Theorem rankcf 8415
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 7483 . . 3  |-  ( rank `  A )  e.  On
2 onzsl 4653 . . 3  |-  ( (
rank `  A )  e.  On  <->  ( ( rank `  A )  =  (/)  \/ 
E. x  e.  On  ( rank `  A )  =  suc  x  \/  (
( rank `  A )  e.  _V  /\  Lim  ( rank `  A ) ) ) )
31, 2mpbi 199 . 2  |-  ( (
rank `  A )  =  (/)  \/  E. x  e.  On  ( rank `  A
)  =  suc  x  \/  ( ( rank `  A
)  e.  _V  /\  Lim  ( rank `  A
) ) )
4 sdom0 7009 . . . 4  |-  -.  A  ~< 
(/)
5 fveq2 5541 . . . . . 6  |-  ( (
rank `  A )  =  (/)  ->  ( cf `  ( rank `  A
) )  =  ( cf `  (/) ) )
6 cf0 7893 . . . . . 6  |-  ( cf `  (/) )  =  (/)
75, 6syl6eq 2344 . . . . 5  |-  ( (
rank `  A )  =  (/)  ->  ( cf `  ( rank `  A
) )  =  (/) )
87breq2d 4051 . . . 4  |-  ( (
rank `  A )  =  (/)  ->  ( A  ~<  ( cf `  ( rank `  A ) )  <-> 
A  ~<  (/) ) )
94, 8mtbiri 294 . . 3  |-  ( (
rank `  A )  =  (/)  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
10 fveq2 5541 . . . . . . 7  |-  ( (
rank `  A )  =  suc  x  ->  ( cf `  ( rank `  A
) )  =  ( cf `  suc  x
) )
11 cfsuc 7899 . . . . . . 7  |-  ( x  e.  On  ->  ( cf `  suc  x )  =  1o )
1210, 11sylan9eqr 2350 . . . . . 6  |-  ( ( x  e.  On  /\  ( rank `  A )  =  suc  x )  -> 
( cf `  ( rank `  A ) )  =  1o )
13 nsuceq0 4488 . . . . . . . . 9  |-  suc  x  =/=  (/)
14 neeq1 2467 . . . . . . . . 9  |-  ( (
rank `  A )  =  suc  x  ->  (
( rank `  A )  =/=  (/)  <->  suc  x  =/=  (/) ) )
1513, 14mpbiri 224 . . . . . . . 8  |-  ( (
rank `  A )  =  suc  x  ->  ( rank `  A )  =/=  (/) )
16 fveq2 5541 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  ( rank `  A )  =  (
rank `  (/) ) )
17 0elon 4461 . . . . . . . . . . . . 13  |-  (/)  e.  On
18 r1fnon 7455 . . . . . . . . . . . . . 14  |-  R1  Fn  On
19 fndm 5359 . . . . . . . . . . . . . 14  |-  ( R1  Fn  On  ->  dom  R1  =  On )
2018, 19ax-mp 8 . . . . . . . . . . . . 13  |-  dom  R1  =  On
2117, 20eleqtrri 2369 . . . . . . . . . . . 12  |-  (/)  e.  dom  R1
22 rankonid 7517 . . . . . . . . . . . 12  |-  ( (/)  e.  dom  R1  <->  ( rank `  (/) )  =  (/) )
2321, 22mpbi 199 . . . . . . . . . . 11  |-  ( rank `  (/) )  =  (/)
2416, 23syl6eq 2344 . . . . . . . . . 10  |-  ( A  =  (/)  ->  ( rank `  A )  =  (/) )
2524necon3i 2498 . . . . . . . . 9  |-  ( (
rank `  A )  =/=  (/)  ->  A  =/=  (/) )
26 rankvaln 7487 . . . . . . . . . . 11  |-  ( -.  A  e.  U. ( R1 " On )  -> 
( rank `  A )  =  (/) )
2726necon1ai 2501 . . . . . . . . . 10  |-  ( (
rank `  A )  =/=  (/)  ->  A  e.  U. ( R1 " On ) )
28 breq2 4043 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( 1o 
~<_  y  <->  1o  ~<_  A )
)
29 neeq1 2467 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
y  =/=  (/)  <->  A  =/=  (/) ) )
30 0sdom1dom 7076 . . . . . . . . . . . 12  |-  ( (/)  ~< 
y  <->  1o  ~<_  y )
31 vex 2804 . . . . . . . . . . . . 13  |-  y  e. 
_V
32310sdom 7008 . . . . . . . . . . . 12  |-  ( (/)  ~< 
y  <->  y  =/=  (/) )
3330, 32bitr3i 242 . . . . . . . . . . 11  |-  ( 1o  ~<_  y  <->  y  =/=  (/) )
3428, 29, 33vtoclbg 2857 . . . . . . . . . 10  |-  ( A  e.  U. ( R1
" On )  -> 
( 1o  ~<_  A  <->  A  =/=  (/) ) )
3527, 34syl 15 . . . . . . . . 9  |-  ( (
rank `  A )  =/=  (/)  ->  ( 1o  ~<_  A 
<->  A  =/=  (/) ) )
3625, 35mpbird 223 . . . . . . . 8  |-  ( (
rank `  A )  =/=  (/)  ->  1o  ~<_  A )
3715, 36syl 15 . . . . . . 7  |-  ( (
rank `  A )  =  suc  x  ->  1o  ~<_  A )
3837adantl 452 . . . . . 6  |-  ( ( x  e.  On  /\  ( rank `  A )  =  suc  x )  ->  1o 
~<_  A )
3912, 38eqbrtrd 4059 . . . . 5  |-  ( ( x  e.  On  /\  ( rank `  A )  =  suc  x )  -> 
( cf `  ( rank `  A ) )  ~<_  A )
4039rexlimiva 2675 . . . 4  |-  ( E. x  e.  On  ( rank `  A )  =  suc  x  ->  ( cf `  ( rank `  A
) )  ~<_  A )
41 domnsym 7003 . . . 4  |-  ( ( cf `  ( rank `  A ) )  ~<_  A  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
4240, 41syl 15 . . 3  |-  ( E. x  e.  On  ( rank `  A )  =  suc  x  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
43 nlim0 4466 . . . . . . . . . . . . . . . . 17  |-  -.  Lim  (/)
44 limeq 4420 . . . . . . . . . . . . . . . . 17  |-  ( (
rank `  A )  =  (/)  ->  ( Lim  ( rank `  A )  <->  Lim  (/) ) )
4543, 44mtbiri 294 . . . . . . . . . . . . . . . 16  |-  ( (
rank `  A )  =  (/)  ->  -.  Lim  ( rank `  A ) )
4626, 45syl 15 . . . . . . . . . . . . . . 15  |-  ( -.  A  e.  U. ( R1 " On )  ->  -.  Lim  ( rank `  A
) )
4746con4i 122 . . . . . . . . . . . . . 14  |-  ( Lim  ( rank `  A
)  ->  A  e.  U. ( R1 " On ) )
48 r1elssi 7493 . . . . . . . . . . . . . 14  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  U. ( R1
" On ) )
4947, 48syl 15 . . . . . . . . . . . . 13  |-  ( Lim  ( rank `  A
)  ->  A  C_  U. ( R1 " On ) )
5049sselda 3193 . . . . . . . . . . . 12  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  x  e.  U. ( R1 " On ) )
51 ranksnb 7515 . . . . . . . . . . . 12  |-  ( x  e.  U. ( R1
" On )  -> 
( rank `  { x } )  =  suc  ( rank `  x )
)
5250, 51syl 15 . . . . . . . . . . 11  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  ( rank `  { x }
)  =  suc  ( rank `  x ) )
53 rankelb 7512 . . . . . . . . . . . . . 14  |-  ( A  e.  U. ( R1
" On )  -> 
( x  e.  A  ->  ( rank `  x
)  e.  ( rank `  A ) ) )
5447, 53syl 15 . . . . . . . . . . . . 13  |-  ( Lim  ( rank `  A
)  ->  ( x  e.  A  ->  ( rank `  x )  e.  (
rank `  A )
) )
55 limsuc 4656 . . . . . . . . . . . . 13  |-  ( Lim  ( rank `  A
)  ->  ( ( rank `  x )  e.  ( rank `  A
)  <->  suc  ( rank `  x
)  e.  ( rank `  A ) ) )
5654, 55sylibd 205 . . . . . . . . . . . 12  |-  ( Lim  ( rank `  A
)  ->  ( x  e.  A  ->  suc  ( rank `  x )  e.  ( rank `  A
) ) )
5756imp 418 . . . . . . . . . . 11  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  suc  ( rank `  x )  e.  ( rank `  A
) )
5852, 57eqeltrd 2370 . . . . . . . . . 10  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  ( rank `  { x }
)  e.  ( rank `  A ) )
59 eleq1a 2365 . . . . . . . . . 10  |-  ( (
rank `  { x } )  e.  (
rank `  A )  ->  ( w  =  (
rank `  { x } )  ->  w  e.  ( rank `  A
) ) )
6058, 59syl 15 . . . . . . . . 9  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  (
w  =  ( rank `  { x } )  ->  w  e.  (
rank `  A )
) )
6160rexlimdva 2680 . . . . . . . 8  |-  ( Lim  ( rank `  A
)  ->  ( E. x  e.  A  w  =  ( rank `  {
x } )  ->  w  e.  ( rank `  A ) ) )
6261abssdv 3260 . . . . . . 7  |-  ( Lim  ( rank `  A
)  ->  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  C_  ( rank `  A ) )
63 snex 4232 . . . . . . . . . . . . 13  |-  { x }  e.  _V
6463dfiun2 3953 . . . . . . . . . . . 12  |-  U_ x  e.  A  { x }  =  U. { y  |  E. x  e.  A  y  =  {
x } }
65 iunid 3973 . . . . . . . . . . . 12  |-  U_ x  e.  A  { x }  =  A
6664, 65eqtr3i 2318 . . . . . . . . . . 11  |-  U. {
y  |  E. x  e.  A  y  =  { x } }  =  A
6766fveq2i 5544 . . . . . . . . . 10  |-  ( rank `  U. { y  |  E. x  e.  A  y  =  { x } } )  =  (
rank `  A )
6848sselda 3193 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  U. ( R1 " On )  /\  x  e.  A )  ->  x  e.  U. ( R1 " On ) )
69 snwf 7497 . . . . . . . . . . . . . . 15  |-  ( x  e.  U. ( R1
" On )  ->  { x }  e.  U. ( R1 " On ) )
70 eleq1a 2365 . . . . . . . . . . . . . . 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 2680 . . . . . . . . . . . . 13  |-  ( A  e.  U. ( R1
" On )  -> 
( E. x  e.  A  y  =  {
x }  ->  y  e.  U. ( R1 " On ) ) )
7372abssdv 3260 . . . . . . . . . . . 12  |-  ( A  e.  U. ( R1
" On )  ->  { y  |  E. x  e.  A  y  =  { x } }  C_ 
U. ( R1 " On ) )
74 abrexexg 5780 . . . . . . . . . . . . 13  |-  ( A  e.  U. ( R1
" On )  ->  { y  |  E. x  e.  A  y  =  { x } }  e.  _V )
75 eleq1 2356 . . . . . . . . . . . . . 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 3212 . . . . . . . . . . . . . 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 2804 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
7877r1elss 7494 . . . . . . . . . . . . . 14  |-  ( z  e.  U. ( R1
" On )  <->  z  C_  U. ( R1 " On ) )
7975, 76, 78vtoclbg 2857 . . . . . . . . . . . . 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 15 . . . . . . . . . . . 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 223 . . . . . . . . . . 11  |-  ( A  e.  U. ( R1
" On )  ->  { y  |  E. x  e.  A  y  =  { x } }  e.  U. ( R1 " On ) )
82 rankuni2b 7541 . . . . . . . . . . 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 15 . . . . . . . . . 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 2342 . . . . . . . . 9  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  U_ z  e.  {
y  |  E. x  e.  A  y  =  { x } } 
( rank `  z )
)
85 fvex 5555 . . . . . . . . . . 11  |-  ( rank `  z )  e.  _V
8685dfiun2 3953 . . . . . . . . . 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 5541 . . . . . . . . . . . 12  |-  ( z  =  { x }  ->  ( rank `  z
)  =  ( rank `  { x } ) )
8863, 87abrexco 5782 . . . . . . . . . . 11  |-  { w  |  E. z  e.  {
y  |  E. x  e.  A  y  =  { x } }
w  =  ( rank `  z ) }  =  { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }
8988unieqi 3853 . . . . . . . . . 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 2316 . . . . . . . . 9  |-  U_ z  e.  { y  |  E. x  e.  A  y  =  { x } } 
( rank `  z )  =  U. { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }
9184, 90syl6req 2345 . . . . . . . 8  |-  ( A  e.  U. ( R1
" On )  ->  U. { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }  =  ( rank `  A
) )
9247, 91syl 15 . . . . . . 7  |-  ( Lim  ( rank `  A
)  ->  U. { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  =  ( rank `  A ) )
93 fvex 5555 . . . . . . . 8  |-  ( rank `  A )  e.  _V
9493cfslb 7908 . . . . . . 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 1279 . . . . . 6  |-  ( Lim  ( rank `  A
)  ->  ( cf `  ( rank `  A
) )  ~<_  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) } )
96 fveq2 5541 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( rank `  y )  =  ( rank `  A
) )
9796fveq2d 5545 . . . . . . . . . 10  |-  ( y  =  A  ->  ( cf `  ( rank `  y
) )  =  ( cf `  ( rank `  A ) ) )
98 breq12 4044 . . . . . . . . . 10  |-  ( ( y  =  A  /\  ( cf `  ( rank `  y ) )  =  ( cf `  ( rank `  A ) ) )  ->  ( y  ~<  ( cf `  ( rank `  y ) )  <-> 
A  ~<  ( cf `  ( rank `  A ) ) ) )
9997, 98mpdan 649 . . . . . . . . 9  |-  ( y  =  A  ->  (
y  ~<  ( cf `  ( rank `  y ) )  <-> 
A  ~<  ( cf `  ( rank `  A ) ) ) )
100 rexeq 2750 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( E. x  e.  y  w  =  ( rank `  { x } )  <->  E. x  e.  A  w  =  ( rank `  { x } ) ) )
101100abbidv 2410 . . . . . . . . . 10  |-  ( y  =  A  ->  { w  |  E. x  e.  y  w  =  ( rank `  { x } ) }  =  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) } )
102 breq12 4044 . . . . . . . . . 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 650 . . . . . . . . 9  |-  ( y  =  A  ->  ( { w  |  E. x  e.  y  w  =  ( rank `  {
x } ) }  ~<_  y  <->  { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }  ~<_  A ) )
10499, 103imbi12d 311 . . . . . . . 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 2296 . . . . . . . . . 10  |-  ( x  e.  y  |->  ( rank `  { x } ) )  =  ( x  e.  y  |->  ( rank `  { x } ) )
106105rnmpt 4941 . . . . . . . . 9  |-  ran  (
x  e.  y  |->  (
rank `  { x } ) )  =  { w  |  E. x  e.  y  w  =  ( rank `  {
x } ) }
107 cfon 7897 . . . . . . . . . . 11  |-  ( cf `  ( rank `  y
) )  e.  On
108 sdomdom 6905 . . . . . . . . . . 11  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  y  ~<_  ( cf `  ( rank `  y
) ) )
109 ondomen 7680 . . . . . . . . . . 11  |-  ( ( ( cf `  ( rank `  y ) )  e.  On  /\  y  ~<_  ( cf `  ( rank `  y ) ) )  ->  y  e.  dom  card )
110107, 108, 109sylancr 644 . . . . . . . . . 10  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  y  e.  dom  card )
111 fvex 5555 . . . . . . . . . . . 12  |-  ( rank `  { x } )  e.  _V
112111, 105fnmpti 5388 . . . . . . . . . . 11  |-  ( x  e.  y  |->  ( rank `  { x } ) )  Fn  y
113 dffn4 5473 . . . . . . . . . . 11  |-  ( ( x  e.  y  |->  (
rank `  { x } ) )  Fn  y  <->  ( x  e.  y  |->  ( rank `  {
x } ) ) : y -onto-> ran  (
x  e.  y  |->  (
rank `  { x } ) ) )
114112, 113mpbi 199 . . . . . . . . . 10  |-  ( x  e.  y  |->  ( rank `  { x } ) ) : y -onto-> ran  ( x  e.  y 
|->  ( rank `  {
x } ) )
115 fodomnum 7700 . . . . . . . . . 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, 115ee10 1366 . . . . . . . . 9  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  ran  ( x  e.  y  |->  ( rank `  { x } ) )  ~<_  y )
117106, 116syl5eqbrr 4073 . . . . . . . 8  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  { w  |  E. x  e.  y  w  =  ( rank `  { x } ) }  ~<_  y )
118104, 117vtoclg 2856 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  -> 
( A  ~<  ( cf `  ( rank `  A
) )  ->  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  ~<_  A ) )
11947, 118syl 15 . . . . . 6  |-  ( Lim  ( rank `  A
)  ->  ( A  ~<  ( cf `  ( rank `  A ) )  ->  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  ~<_  A ) )
120 domtr 6930 . . . . . . 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 15 . . . . . 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, 121ee12an 1353 . . . . 5  |-  ( Lim  ( rank `  A
)  ->  ( A  ~<  ( cf `  ( rank `  A ) )  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) ) )
123122pm2.01d 161 . . . 4  |-  ( Lim  ( rank `  A
)  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
124123adantl 452 . . 3  |-  ( ( ( rank `  A
)  e.  _V  /\  Lim  ( rank `  A
) )  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
1259, 42, 1243jaoi 1245 . 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 8 1  |-  -.  A  ~<  ( cf `  ( rank `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    \/ w3o 933    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   E.wrex 2557   _Vcvv 2801    C_ wss 3165   (/)c0 3468   {csn 3653   U.cuni 3843   U_ciun 3921   class class class wbr 4039    e. cmpt 4093   Oncon0 4408   Lim wlim 4409   suc csuc 4410   dom cdm 4705   ran crn 4706   "cima 4708    Fn wfn 5266   -onto->wfo 5269   ` cfv 5271   1oc1o 6488    ~<_ cdom 6877    ~< csdm 6878   R1cr1 7450   rankcrnk 7451   cardccrd 7584   cfccf 7586
This theorem is referenced by:  inatsk  8416  grur1  8458
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-r1 7452  df-rank 7453  df-card 7588  df-cf 7590  df-acn 7591
  Copyright terms: Public domain W3C validator