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Theorem rankcf 8652
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 7721 . . 3  |-  ( rank `  A )  e.  On
2 onzsl 4826 . . 3  |-  ( (
rank `  A )  e.  On  <->  ( ( rank `  A )  =  (/)  \/ 
E. x  e.  On  ( rank `  A )  =  suc  x  \/  (
( rank `  A )  e.  _V  /\  Lim  ( rank `  A ) ) ) )
31, 2mpbi 200 . 2  |-  ( (
rank `  A )  =  (/)  \/  E. x  e.  On  ( rank `  A
)  =  suc  x  \/  ( ( rank `  A
)  e.  _V  /\  Lim  ( rank `  A
) ) )
4 sdom0 7239 . . . 4  |-  -.  A  ~< 
(/)
5 fveq2 5728 . . . . . 6  |-  ( (
rank `  A )  =  (/)  ->  ( cf `  ( rank `  A
) )  =  ( cf `  (/) ) )
6 cf0 8131 . . . . . 6  |-  ( cf `  (/) )  =  (/)
75, 6syl6eq 2484 . . . . 5  |-  ( (
rank `  A )  =  (/)  ->  ( cf `  ( rank `  A
) )  =  (/) )
87breq2d 4224 . . . 4  |-  ( (
rank `  A )  =  (/)  ->  ( A  ~<  ( cf `  ( rank `  A ) )  <-> 
A  ~<  (/) ) )
94, 8mtbiri 295 . . 3  |-  ( (
rank `  A )  =  (/)  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
10 fveq2 5728 . . . . . . 7  |-  ( (
rank `  A )  =  suc  x  ->  ( cf `  ( rank `  A
) )  =  ( cf `  suc  x
) )
11 cfsuc 8137 . . . . . . 7  |-  ( x  e.  On  ->  ( cf `  suc  x )  =  1o )
1210, 11sylan9eqr 2490 . . . . . 6  |-  ( ( x  e.  On  /\  ( rank `  A )  =  suc  x )  -> 
( cf `  ( rank `  A ) )  =  1o )
13 nsuceq0 4661 . . . . . . . . 9  |-  suc  x  =/=  (/)
14 neeq1 2609 . . . . . . . . 9  |-  ( (
rank `  A )  =  suc  x  ->  (
( rank `  A )  =/=  (/)  <->  suc  x  =/=  (/) ) )
1513, 14mpbiri 225 . . . . . . . 8  |-  ( (
rank `  A )  =  suc  x  ->  ( rank `  A )  =/=  (/) )
16 fveq2 5728 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  ( rank `  A )  =  (
rank `  (/) ) )
17 0elon 4634 . . . . . . . . . . . . 13  |-  (/)  e.  On
18 r1fnon 7693 . . . . . . . . . . . . . 14  |-  R1  Fn  On
19 fndm 5544 . . . . . . . . . . . . . 14  |-  ( R1  Fn  On  ->  dom  R1  =  On )
2018, 19ax-mp 8 . . . . . . . . . . . . 13  |-  dom  R1  =  On
2117, 20eleqtrri 2509 . . . . . . . . . . . 12  |-  (/)  e.  dom  R1
22 rankonid 7755 . . . . . . . . . . . 12  |-  ( (/)  e.  dom  R1  <->  ( rank `  (/) )  =  (/) )
2321, 22mpbi 200 . . . . . . . . . . 11  |-  ( rank `  (/) )  =  (/)
2416, 23syl6eq 2484 . . . . . . . . . 10  |-  ( A  =  (/)  ->  ( rank `  A )  =  (/) )
2524necon3i 2643 . . . . . . . . 9  |-  ( (
rank `  A )  =/=  (/)  ->  A  =/=  (/) )
26 rankvaln 7725 . . . . . . . . . . 11  |-  ( -.  A  e.  U. ( R1 " On )  -> 
( rank `  A )  =  (/) )
2726necon1ai 2646 . . . . . . . . . 10  |-  ( (
rank `  A )  =/=  (/)  ->  A  e.  U. ( R1 " On ) )
28 breq2 4216 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( 1o 
~<_  y  <->  1o  ~<_  A )
)
29 neeq1 2609 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
y  =/=  (/)  <->  A  =/=  (/) ) )
30 0sdom1dom 7306 . . . . . . . . . . . 12  |-  ( (/)  ~< 
y  <->  1o  ~<_  y )
31 vex 2959 . . . . . . . . . . . . 13  |-  y  e. 
_V
32310sdom 7238 . . . . . . . . . . . 12  |-  ( (/)  ~< 
y  <->  y  =/=  (/) )
3330, 32bitr3i 243 . . . . . . . . . . 11  |-  ( 1o  ~<_  y  <->  y  =/=  (/) )
3428, 29, 33vtoclbg 3012 . . . . . . . . . 10  |-  ( A  e.  U. ( R1
" On )  -> 
( 1o  ~<_  A  <->  A  =/=  (/) ) )
3527, 34syl 16 . . . . . . . . 9  |-  ( (
rank `  A )  =/=  (/)  ->  ( 1o  ~<_  A 
<->  A  =/=  (/) ) )
3625, 35mpbird 224 . . . . . . . 8  |-  ( (
rank `  A )  =/=  (/)  ->  1o  ~<_  A )
3715, 36syl 16 . . . . . . 7  |-  ( (
rank `  A )  =  suc  x  ->  1o  ~<_  A )
3837adantl 453 . . . . . 6  |-  ( ( x  e.  On  /\  ( rank `  A )  =  suc  x )  ->  1o 
~<_  A )
3912, 38eqbrtrd 4232 . . . . 5  |-  ( ( x  e.  On  /\  ( rank `  A )  =  suc  x )  -> 
( cf `  ( rank `  A ) )  ~<_  A )
4039rexlimiva 2825 . . . 4  |-  ( E. x  e.  On  ( rank `  A )  =  suc  x  ->  ( cf `  ( rank `  A
) )  ~<_  A )
41 domnsym 7233 . . . 4  |-  ( ( cf `  ( rank `  A ) )  ~<_  A  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
4240, 41syl 16 . . 3  |-  ( E. x  e.  On  ( rank `  A )  =  suc  x  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
43 nlim0 4639 . . . . . . . . . . . . . . . . 17  |-  -.  Lim  (/)
44 limeq 4593 . . . . . . . . . . . . . . . . 17  |-  ( (
rank `  A )  =  (/)  ->  ( Lim  ( rank `  A )  <->  Lim  (/) ) )
4543, 44mtbiri 295 . . . . . . . . . . . . . . . 16  |-  ( (
rank `  A )  =  (/)  ->  -.  Lim  ( rank `  A ) )
4626, 45syl 16 . . . . . . . . . . . . . . 15  |-  ( -.  A  e.  U. ( R1 " On )  ->  -.  Lim  ( rank `  A
) )
4746con4i 124 . . . . . . . . . . . . . 14  |-  ( Lim  ( rank `  A
)  ->  A  e.  U. ( R1 " On ) )
48 r1elssi 7731 . . . . . . . . . . . . . 14  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  U. ( R1
" On ) )
4947, 48syl 16 . . . . . . . . . . . . 13  |-  ( Lim  ( rank `  A
)  ->  A  C_  U. ( R1 " On ) )
5049sselda 3348 . . . . . . . . . . . 12  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  x  e.  U. ( R1 " On ) )
51 ranksnb 7753 . . . . . . . . . . . 12  |-  ( x  e.  U. ( R1
" On )  -> 
( rank `  { x } )  =  suc  ( rank `  x )
)
5250, 51syl 16 . . . . . . . . . . 11  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  ( rank `  { x }
)  =  suc  ( rank `  x ) )
53 rankelb 7750 . . . . . . . . . . . . . 14  |-  ( A  e.  U. ( R1
" On )  -> 
( x  e.  A  ->  ( rank `  x
)  e.  ( rank `  A ) ) )
5447, 53syl 16 . . . . . . . . . . . . 13  |-  ( Lim  ( rank `  A
)  ->  ( x  e.  A  ->  ( rank `  x )  e.  (
rank `  A )
) )
55 limsuc 4829 . . . . . . . . . . . . 13  |-  ( Lim  ( rank `  A
)  ->  ( ( rank `  x )  e.  ( rank `  A
)  <->  suc  ( rank `  x
)  e.  ( rank `  A ) ) )
5654, 55sylibd 206 . . . . . . . . . . . 12  |-  ( Lim  ( rank `  A
)  ->  ( x  e.  A  ->  suc  ( rank `  x )  e.  ( rank `  A
) ) )
5756imp 419 . . . . . . . . . . 11  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  suc  ( rank `  x )  e.  ( rank `  A
) )
5852, 57eqeltrd 2510 . . . . . . . . . 10  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  ( rank `  { x }
)  e.  ( rank `  A ) )
59 eleq1a 2505 . . . . . . . . . 10  |-  ( (
rank `  { x } )  e.  (
rank `  A )  ->  ( w  =  (
rank `  { x } )  ->  w  e.  ( rank `  A
) ) )
6058, 59syl 16 . . . . . . . . 9  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  (
w  =  ( rank `  { x } )  ->  w  e.  (
rank `  A )
) )
6160rexlimdva 2830 . . . . . . . 8  |-  ( Lim  ( rank `  A
)  ->  ( E. x  e.  A  w  =  ( rank `  {
x } )  ->  w  e.  ( rank `  A ) ) )
6261abssdv 3417 . . . . . . 7  |-  ( Lim  ( rank `  A
)  ->  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  C_  ( rank `  A ) )
63 snex 4405 . . . . . . . . . . . . 13  |-  { x }  e.  _V
6463dfiun2 4125 . . . . . . . . . . . 12  |-  U_ x  e.  A  { x }  =  U. { y  |  E. x  e.  A  y  =  {
x } }
65 iunid 4146 . . . . . . . . . . . 12  |-  U_ x  e.  A  { x }  =  A
6664, 65eqtr3i 2458 . . . . . . . . . . 11  |-  U. {
y  |  E. x  e.  A  y  =  { x } }  =  A
6766fveq2i 5731 . . . . . . . . . 10  |-  ( rank `  U. { y  |  E. x  e.  A  y  =  { x } } )  =  (
rank `  A )
6848sselda 3348 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  U. ( R1 " On )  /\  x  e.  A )  ->  x  e.  U. ( R1 " On ) )
69 snwf 7735 . . . . . . . . . . . . . . 15  |-  ( x  e.  U. ( R1
" On )  ->  { x }  e.  U. ( R1 " On ) )
70 eleq1a 2505 . . . . . . . . . . . . . . 15  |-  ( { x }  e.  U. ( R1 " On )  ->  ( y  =  { x }  ->  y  e.  U. ( R1
" On ) ) )
7168, 69, 703syl 19 . . . . . . . . . . . . . 14  |-  ( ( A  e.  U. ( R1 " On )  /\  x  e.  A )  ->  ( y  =  {
x }  ->  y  e.  U. ( R1 " On ) ) )
7271rexlimdva 2830 . . . . . . . . . . . . 13  |-  ( A  e.  U. ( R1
" On )  -> 
( E. x  e.  A  y  =  {
x }  ->  y  e.  U. ( R1 " On ) ) )
7372abssdv 3417 . . . . . . . . . . . 12  |-  ( A  e.  U. ( R1
" On )  ->  { y  |  E. x  e.  A  y  =  { x } }  C_ 
U. ( R1 " On ) )
74 abrexexg 5984 . . . . . . . . . . . . 13  |-  ( A  e.  U. ( R1
" On )  ->  { y  |  E. x  e.  A  y  =  { x } }  e.  _V )
75 eleq1 2496 . . . . . . . . . . . . . 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 3369 . . . . . . . . . . . . . 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 2959 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
7877r1elss 7732 . . . . . . . . . . . . . 14  |-  ( z  e.  U. ( R1
" On )  <->  z  C_  U. ( R1 " On ) )
7975, 76, 78vtoclbg 3012 . . . . . . . . . . . . 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 16 . . . . . . . . . . . 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 224 . . . . . . . . . . 11  |-  ( A  e.  U. ( R1
" On )  ->  { y  |  E. x  e.  A  y  =  { x } }  e.  U. ( R1 " On ) )
82 rankuni2b 7779 . . . . . . . . . . 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 16 . . . . . . . . . 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 2482 . . . . . . . . 9  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  U_ z  e.  {
y  |  E. x  e.  A  y  =  { x } } 
( rank `  z )
)
85 fvex 5742 . . . . . . . . . . 11  |-  ( rank `  z )  e.  _V
8685dfiun2 4125 . . . . . . . . . 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 5728 . . . . . . . . . . . 12  |-  ( z  =  { x }  ->  ( rank `  z
)  =  ( rank `  { x } ) )
8863, 87abrexco 5986 . . . . . . . . . . 11  |-  { w  |  E. z  e.  {
y  |  E. x  e.  A  y  =  { x } }
w  =  ( rank `  z ) }  =  { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }
8988unieqi 4025 . . . . . . . . . 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 2456 . . . . . . . . 9  |-  U_ z  e.  { y  |  E. x  e.  A  y  =  { x } } 
( rank `  z )  =  U. { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }
9184, 90syl6req 2485 . . . . . . . 8  |-  ( A  e.  U. ( R1
" On )  ->  U. { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }  =  ( rank `  A
) )
9247, 91syl 16 . . . . . . 7  |-  ( Lim  ( rank `  A
)  ->  U. { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  =  ( rank `  A ) )
93 fvex 5742 . . . . . . . 8  |-  ( rank `  A )  e.  _V
9493cfslb 8146 . . . . . . 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 1281 . . . . . 6  |-  ( Lim  ( rank `  A
)  ->  ( cf `  ( rank `  A
) )  ~<_  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) } )
96 fveq2 5728 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( rank `  y )  =  ( rank `  A
) )
9796fveq2d 5732 . . . . . . . . . 10  |-  ( y  =  A  ->  ( cf `  ( rank `  y
) )  =  ( cf `  ( rank `  A ) ) )
98 breq12 4217 . . . . . . . . . 10  |-  ( ( y  =  A  /\  ( cf `  ( rank `  y ) )  =  ( cf `  ( rank `  A ) ) )  ->  ( y  ~<  ( cf `  ( rank `  y ) )  <-> 
A  ~<  ( cf `  ( rank `  A ) ) ) )
9997, 98mpdan 650 . . . . . . . . 9  |-  ( y  =  A  ->  (
y  ~<  ( cf `  ( rank `  y ) )  <-> 
A  ~<  ( cf `  ( rank `  A ) ) ) )
100 rexeq 2905 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( E. x  e.  y  w  =  ( rank `  { x } )  <->  E. x  e.  A  w  =  ( rank `  { x } ) ) )
101100abbidv 2550 . . . . . . . . . 10  |-  ( y  =  A  ->  { w  |  E. x  e.  y  w  =  ( rank `  { x } ) }  =  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) } )
102 breq12 4217 . . . . . . . . . 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 651 . . . . . . . . 9  |-  ( y  =  A  ->  ( { w  |  E. x  e.  y  w  =  ( rank `  {
x } ) }  ~<_  y  <->  { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }  ~<_  A ) )
10499, 103imbi12d 312 . . . . . . . 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 2436 . . . . . . . . . 10  |-  ( x  e.  y  |->  ( rank `  { x } ) )  =  ( x  e.  y  |->  ( rank `  { x } ) )
106105rnmpt 5116 . . . . . . . . 9  |-  ran  (
x  e.  y  |->  (
rank `  { x } ) )  =  { w  |  E. x  e.  y  w  =  ( rank `  {
x } ) }
107 cfon 8135 . . . . . . . . . . 11  |-  ( cf `  ( rank `  y
) )  e.  On
108 sdomdom 7135 . . . . . . . . . . 11  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  y  ~<_  ( cf `  ( rank `  y
) ) )
109 ondomen 7918 . . . . . . . . . . 11  |-  ( ( ( cf `  ( rank `  y ) )  e.  On  /\  y  ~<_  ( cf `  ( rank `  y ) ) )  ->  y  e.  dom  card )
110107, 108, 109sylancr 645 . . . . . . . . . 10  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  y  e.  dom  card )
111 fvex 5742 . . . . . . . . . . . 12  |-  ( rank `  { x } )  e.  _V
112111, 105fnmpti 5573 . . . . . . . . . . 11  |-  ( x  e.  y  |->  ( rank `  { x } ) )  Fn  y
113 dffn4 5659 . . . . . . . . . . 11  |-  ( ( x  e.  y  |->  (
rank `  { x } ) )  Fn  y  <->  ( x  e.  y  |->  ( rank `  {
x } ) ) : y -onto-> ran  (
x  e.  y  |->  (
rank `  { x } ) ) )
114112, 113mpbi 200 . . . . . . . . . 10  |-  ( x  e.  y  |->  ( rank `  { x } ) ) : y -onto-> ran  ( x  e.  y 
|->  ( rank `  {
x } ) )
115 fodomnum 7938 . . . . . . . . . 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 1385 . . . . . . . . 9  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  ran  ( x  e.  y  |->  ( rank `  { x } ) )  ~<_  y )
117106, 116syl5eqbrr 4246 . . . . . . . 8  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  { w  |  E. x  e.  y  w  =  ( rank `  { x } ) }  ~<_  y )
118104, 117vtoclg 3011 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  -> 
( A  ~<  ( cf `  ( rank `  A
) )  ->  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  ~<_  A ) )
11947, 118syl 16 . . . . . 6  |-  ( Lim  ( rank `  A
)  ->  ( A  ~<  ( cf `  ( rank `  A ) )  ->  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  ~<_  A ) )
120 domtr 7160 . . . . . . 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 16 . . . . . 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 1372 . . . . 5  |-  ( Lim  ( rank `  A
)  ->  ( A  ~<  ( cf `  ( rank `  A ) )  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) ) )
123122pm2.01d 163 . . . 4  |-  ( Lim  ( rank `  A
)  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
124123adantl 453 . . 3  |-  ( ( ( rank `  A
)  e.  _V  /\  Lim  ( rank `  A
) )  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
1259, 42, 1243jaoi 1247 . 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 177    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599   E.wrex 2706   _Vcvv 2956    C_ wss 3320   (/)c0 3628   {csn 3814   U.cuni 4015   U_ciun 4093   class class class wbr 4212    e. cmpt 4266   Oncon0 4581   Lim wlim 4582   suc csuc 4583   dom cdm 4878   ran crn 4879   "cima 4881    Fn wfn 5449   -onto->wfo 5452   ` cfv 5454   1oc1o 6717    ~<_ cdom 7107    ~< csdm 7108   R1cr1 7688   rankcrnk 7689   cardccrd 7822   cfccf 7824
This theorem is referenced by:  inatsk  8653  grur1  8695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-r1 7690  df-rank 7691  df-card 7826  df-cf 7828  df-acn 7829
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