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Theorem rankcf 9154
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 8212 . . 3  |-  ( rank `  A )  e.  On
2 onzsl 6660 . . 3  |-  ( (
rank `  A )  e.  On  <->  ( ( rank `  A )  =  (/)  \/ 
E. x  e.  On  ( rank `  A )  =  suc  x  \/  (
( rank `  A )  e.  _V  /\  Lim  ( rank `  A ) ) ) )
31, 2mpbi 208 . 2  |-  ( (
rank `  A )  =  (/)  \/  E. x  e.  On  ( rank `  A
)  =  suc  x  \/  ( ( rank `  A
)  e.  _V  /\  Lim  ( rank `  A
) ) )
4 sdom0 7649 . . . 4  |-  -.  A  ~< 
(/)
5 fveq2 5865 . . . . . 6  |-  ( (
rank `  A )  =  (/)  ->  ( cf `  ( rank `  A
) )  =  ( cf `  (/) ) )
6 cf0 8630 . . . . . 6  |-  ( cf `  (/) )  =  (/)
75, 6syl6eq 2524 . . . . 5  |-  ( (
rank `  A )  =  (/)  ->  ( cf `  ( rank `  A
) )  =  (/) )
87breq2d 4459 . . . 4  |-  ( (
rank `  A )  =  (/)  ->  ( A  ~<  ( cf `  ( rank `  A ) )  <-> 
A  ~<  (/) ) )
94, 8mtbiri 303 . . 3  |-  ( (
rank `  A )  =  (/)  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
10 fveq2 5865 . . . . . . 7  |-  ( (
rank `  A )  =  suc  x  ->  ( cf `  ( rank `  A
) )  =  ( cf `  suc  x
) )
11 cfsuc 8636 . . . . . . 7  |-  ( x  e.  On  ->  ( cf `  suc  x )  =  1o )
1210, 11sylan9eqr 2530 . . . . . 6  |-  ( ( x  e.  On  /\  ( rank `  A )  =  suc  x )  -> 
( cf `  ( rank `  A ) )  =  1o )
13 nsuceq0 4958 . . . . . . . . 9  |-  suc  x  =/=  (/)
14 neeq1 2748 . . . . . . . . 9  |-  ( (
rank `  A )  =  suc  x  ->  (
( rank `  A )  =/=  (/)  <->  suc  x  =/=  (/) ) )
1513, 14mpbiri 233 . . . . . . . 8  |-  ( (
rank `  A )  =  suc  x  ->  ( rank `  A )  =/=  (/) )
16 fveq2 5865 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  ( rank `  A )  =  (
rank `  (/) ) )
17 0elon 4931 . . . . . . . . . . . . 13  |-  (/)  e.  On
18 r1fnon 8184 . . . . . . . . . . . . . 14  |-  R1  Fn  On
19 fndm 5679 . . . . . . . . . . . . . 14  |-  ( R1  Fn  On  ->  dom  R1  =  On )
2018, 19ax-mp 5 . . . . . . . . . . . . 13  |-  dom  R1  =  On
2117, 20eleqtrri 2554 . . . . . . . . . . . 12  |-  (/)  e.  dom  R1
22 rankonid 8246 . . . . . . . . . . . 12  |-  ( (/)  e.  dom  R1  <->  ( rank `  (/) )  =  (/) )
2321, 22mpbi 208 . . . . . . . . . . 11  |-  ( rank `  (/) )  =  (/)
2416, 23syl6eq 2524 . . . . . . . . . 10  |-  ( A  =  (/)  ->  ( rank `  A )  =  (/) )
2524necon3i 2707 . . . . . . . . 9  |-  ( (
rank `  A )  =/=  (/)  ->  A  =/=  (/) )
26 rankvaln 8216 . . . . . . . . . . 11  |-  ( -.  A  e.  U. ( R1 " On )  -> 
( rank `  A )  =  (/) )
2726necon1ai 2698 . . . . . . . . . 10  |-  ( (
rank `  A )  =/=  (/)  ->  A  e.  U. ( R1 " On ) )
28 breq2 4451 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( 1o 
~<_  y  <->  1o  ~<_  A )
)
29 neeq1 2748 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
y  =/=  (/)  <->  A  =/=  (/) ) )
30 0sdom1dom 7717 . . . . . . . . . . . 12  |-  ( (/)  ~< 
y  <->  1o  ~<_  y )
31 vex 3116 . . . . . . . . . . . . 13  |-  y  e. 
_V
32310sdom 7648 . . . . . . . . . . . 12  |-  ( (/)  ~< 
y  <->  y  =/=  (/) )
3330, 32bitr3i 251 . . . . . . . . . . 11  |-  ( 1o  ~<_  y  <->  y  =/=  (/) )
3428, 29, 33vtoclbg 3172 . . . . . . . . . 10  |-  ( A  e.  U. ( R1
" On )  -> 
( 1o  ~<_  A  <->  A  =/=  (/) ) )
3527, 34syl 16 . . . . . . . . 9  |-  ( (
rank `  A )  =/=  (/)  ->  ( 1o  ~<_  A 
<->  A  =/=  (/) ) )
3625, 35mpbird 232 . . . . . . . 8  |-  ( (
rank `  A )  =/=  (/)  ->  1o  ~<_  A )
3715, 36syl 16 . . . . . . 7  |-  ( (
rank `  A )  =  suc  x  ->  1o  ~<_  A )
3837adantl 466 . . . . . 6  |-  ( ( x  e.  On  /\  ( rank `  A )  =  suc  x )  ->  1o 
~<_  A )
3912, 38eqbrtrd 4467 . . . . 5  |-  ( ( x  e.  On  /\  ( rank `  A )  =  suc  x )  -> 
( cf `  ( rank `  A ) )  ~<_  A )
4039rexlimiva 2951 . . . 4  |-  ( E. x  e.  On  ( rank `  A )  =  suc  x  ->  ( cf `  ( rank `  A
) )  ~<_  A )
41 domnsym 7643 . . . 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 4936 . . . . . . . . . . . . . . . . 17  |-  -.  Lim  (/)
44 limeq 4890 . . . . . . . . . . . . . . . . 17  |-  ( (
rank `  A )  =  (/)  ->  ( Lim  ( rank `  A )  <->  Lim  (/) ) )
4543, 44mtbiri 303 . . . . . . . . . . . . . . . 16  |-  ( (
rank `  A )  =  (/)  ->  -.  Lim  ( rank `  A ) )
4626, 45syl 16 . . . . . . . . . . . . . . 15  |-  ( -.  A  e.  U. ( R1 " On )  ->  -.  Lim  ( rank `  A
) )
4746con4i 130 . . . . . . . . . . . . . 14  |-  ( Lim  ( rank `  A
)  ->  A  e.  U. ( R1 " On ) )
48 r1elssi 8222 . . . . . . . . . . . . . 14  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  U. ( R1
" On ) )
4947, 48syl 16 . . . . . . . . . . . . 13  |-  ( Lim  ( rank `  A
)  ->  A  C_  U. ( R1 " On ) )
5049sselda 3504 . . . . . . . . . . . 12  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  x  e.  U. ( R1 " On ) )
51 ranksnb 8244 . . . . . . . . . . . 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 8241 . . . . . . . . . . . . . 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 6663 . . . . . . . . . . . . 13  |-  ( Lim  ( rank `  A
)  ->  ( ( rank `  x )  e.  ( rank `  A
)  <->  suc  ( rank `  x
)  e.  ( rank `  A ) ) )
5654, 55sylibd 214 . . . . . . . . . . . 12  |-  ( Lim  ( rank `  A
)  ->  ( x  e.  A  ->  suc  ( rank `  x )  e.  ( rank `  A
) ) )
5756imp 429 . . . . . . . . . . 11  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  suc  ( rank `  x )  e.  ( rank `  A
) )
5852, 57eqeltrd 2555 . . . . . . . . . 10  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  ( rank `  { x }
)  e.  ( rank `  A ) )
59 eleq1a 2550 . . . . . . . . . 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 2955 . . . . . . . 8  |-  ( Lim  ( rank `  A
)  ->  ( E. x  e.  A  w  =  ( rank `  {
x } )  ->  w  e.  ( rank `  A ) ) )
6261abssdv 3574 . . . . . . 7  |-  ( Lim  ( rank `  A
)  ->  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  C_  ( rank `  A ) )
63 snex 4688 . . . . . . . . . . . . 13  |-  { x }  e.  _V
6463dfiun2 4359 . . . . . . . . . . . 12  |-  U_ x  e.  A  { x }  =  U. { y  |  E. x  e.  A  y  =  {
x } }
65 iunid 4380 . . . . . . . . . . . 12  |-  U_ x  e.  A  { x }  =  A
6664, 65eqtr3i 2498 . . . . . . . . . . 11  |-  U. {
y  |  E. x  e.  A  y  =  { x } }  =  A
6766fveq2i 5868 . . . . . . . . . 10  |-  ( rank `  U. { y  |  E. x  e.  A  y  =  { x } } )  =  (
rank `  A )
6848sselda 3504 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  U. ( R1 " On )  /\  x  e.  A )  ->  x  e.  U. ( R1 " On ) )
69 snwf 8226 . . . . . . . . . . . . . . 15  |-  ( x  e.  U. ( R1
" On )  ->  { x }  e.  U. ( R1 " On ) )
70 eleq1a 2550 . . . . . . . . . . . . . . 15  |-  ( { x }  e.  U. ( R1 " On )  ->  ( y  =  { x }  ->  y  e.  U. ( R1
" On ) ) )
7168, 69, 703syl 20 . . . . . . . . . . . . . 14  |-  ( ( A  e.  U. ( R1 " On )  /\  x  e.  A )  ->  ( y  =  {
x }  ->  y  e.  U. ( R1 " On ) ) )
7271rexlimdva 2955 . . . . . . . . . . . . 13  |-  ( A  e.  U. ( R1
" On )  -> 
( E. x  e.  A  y  =  {
x }  ->  y  e.  U. ( R1 " On ) ) )
7372abssdv 3574 . . . . . . . . . . . 12  |-  ( A  e.  U. ( R1
" On )  ->  { y  |  E. x  e.  A  y  =  { x } }  C_ 
U. ( R1 " On ) )
74 abrexexg 6759 . . . . . . . . . . . . 13  |-  ( A  e.  U. ( R1
" On )  ->  { y  |  E. x  e.  A  y  =  { x } }  e.  _V )
75 eleq1 2539 . . . . . . . . . . . . . 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 3525 . . . . . . . . . . . . . 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 3116 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
7877r1elss 8223 . . . . . . . . . . . . . 14  |-  ( z  e.  U. ( R1
" On )  <->  z  C_  U. ( R1 " On ) )
7975, 76, 78vtoclbg 3172 . . . . . . . . . . . . 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 232 . . . . . . . . . . 11  |-  ( A  e.  U. ( R1
" On )  ->  { y  |  E. x  e.  A  y  =  { x } }  e.  U. ( R1 " On ) )
82 rankuni2b 8270 . . . . . . . . . . 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 2522 . . . . . . . . 9  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  U_ z  e.  {
y  |  E. x  e.  A  y  =  { x } } 
( rank `  z )
)
85 fvex 5875 . . . . . . . . . . 11  |-  ( rank `  z )  e.  _V
8685dfiun2 4359 . . . . . . . . . 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 5865 . . . . . . . . . . . 12  |-  ( z  =  { x }  ->  ( rank `  z
)  =  ( rank `  { x } ) )
8863, 87abrexco 6143 . . . . . . . . . . 11  |-  { w  |  E. z  e.  {
y  |  E. x  e.  A  y  =  { x } }
w  =  ( rank `  z ) }  =  { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }
8988unieqi 4254 . . . . . . . . . 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 2496 . . . . . . . . 9  |-  U_ z  e.  { y  |  E. x  e.  A  y  =  { x } } 
( rank `  z )  =  U. { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }
9184, 90syl6req 2525 . . . . . . . 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 5875 . . . . . . . 8  |-  ( rank `  A )  e.  _V
9493cfslb 8645 . . . . . . 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 1326 . . . . . 6  |-  ( Lim  ( rank `  A
)  ->  ( cf `  ( rank `  A
) )  ~<_  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) } )
96 fveq2 5865 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( rank `  y )  =  ( rank `  A
) )
9796fveq2d 5869 . . . . . . . . . 10  |-  ( y  =  A  ->  ( cf `  ( rank `  y
) )  =  ( cf `  ( rank `  A ) ) )
98 breq12 4452 . . . . . . . . . 10  |-  ( ( y  =  A  /\  ( cf `  ( rank `  y ) )  =  ( cf `  ( rank `  A ) ) )  ->  ( y  ~<  ( cf `  ( rank `  y ) )  <-> 
A  ~<  ( cf `  ( rank `  A ) ) ) )
9997, 98mpdan 668 . . . . . . . . 9  |-  ( y  =  A  ->  (
y  ~<  ( cf `  ( rank `  y ) )  <-> 
A  ~<  ( cf `  ( rank `  A ) ) ) )
100 rexeq 3059 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( E. x  e.  y  w  =  ( rank `  { x } )  <->  E. x  e.  A  w  =  ( rank `  { x } ) ) )
101100abbidv 2603 . . . . . . . . . 10  |-  ( y  =  A  ->  { w  |  E. x  e.  y  w  =  ( rank `  { x } ) }  =  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) } )
102 breq12 4452 . . . . . . . . . 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 669 . . . . . . . . 9  |-  ( y  =  A  ->  ( { w  |  E. x  e.  y  w  =  ( rank `  {
x } ) }  ~<_  y  <->  { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }  ~<_  A ) )
10499, 103imbi12d 320 . . . . . . . 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 2467 . . . . . . . . . 10  |-  ( x  e.  y  |->  ( rank `  { x } ) )  =  ( x  e.  y  |->  ( rank `  { x } ) )
106105rnmpt 5247 . . . . . . . . 9  |-  ran  (
x  e.  y  |->  (
rank `  { x } ) )  =  { w  |  E. x  e.  y  w  =  ( rank `  {
x } ) }
107 cfon 8634 . . . . . . . . . . 11  |-  ( cf `  ( rank `  y
) )  e.  On
108 sdomdom 7543 . . . . . . . . . . 11  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  y  ~<_  ( cf `  ( rank `  y
) ) )
109 ondomen 8417 . . . . . . . . . . 11  |-  ( ( ( cf `  ( rank `  y ) )  e.  On  /\  y  ~<_  ( cf `  ( rank `  y ) ) )  ->  y  e.  dom  card )
110107, 108, 109sylancr 663 . . . . . . . . . 10  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  y  e.  dom  card )
111 fvex 5875 . . . . . . . . . . . 12  |-  ( rank `  { x } )  e.  _V
112111, 105fnmpti 5708 . . . . . . . . . . 11  |-  ( x  e.  y  |->  ( rank `  { x } ) )  Fn  y
113 dffn4 5800 . . . . . . . . . . 11  |-  ( ( x  e.  y  |->  (
rank `  { x } ) )  Fn  y  <->  ( x  e.  y  |->  ( rank `  {
x } ) ) : y -onto-> ran  (
x  e.  y  |->  (
rank `  { x } ) ) )
114112, 113mpbi 208 . . . . . . . . . 10  |-  ( x  e.  y  |->  ( rank `  { x } ) ) : y -onto-> ran  ( x  e.  y 
|->  ( rank `  {
x } ) )
115 fodomnum 8437 . . . . . . . . . 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 18 . . . . . . . . 9  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  ran  ( x  e.  y  |->  ( rank `  { x } ) )  ~<_  y )
117106, 116syl5eqbrr 4481 . . . . . . . 8  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  { w  |  E. x  e.  y  w  =  ( rank `  { x } ) }  ~<_  y )
118104, 117vtoclg 3171 . . . . . . 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 7568 . . . . . . 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, 121syl6an 545 . . . . 5  |-  ( Lim  ( rank `  A
)  ->  ( A  ~<  ( cf `  ( rank `  A ) )  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) ) )
123122pm2.01d 169 . . . 4  |-  ( Lim  ( rank `  A
)  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
124123adantl 466 . . 3  |-  ( ( ( rank `  A
)  e.  _V  /\  Lim  ( rank `  A
) )  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
1259, 42, 1243jaoi 1291 . 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 184    /\ wa 369    \/ w3o 972    = wceq 1379    e. wcel 1767   {cab 2452    =/= wne 2662   E.wrex 2815   _Vcvv 3113    C_ wss 3476   (/)c0 3785   {csn 4027   U.cuni 4245   U_ciun 4325   class class class wbr 4447    |-> cmpt 4505   Oncon0 4878   Lim wlim 4879   suc csuc 4880   dom cdm 4999   ran crn 5000   "cima 5002    Fn wfn 5582   -onto->wfo 5585   ` cfv 5587   1oc1o 7123    ~<_ cdom 7514    ~< csdm 7515   R1cr1 8179   rankcrnk 8180   cardccrd 8315   cfccf 8317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-r1 8181  df-rank 8182  df-card 8319  df-cf 8321  df-acn 8322
This theorem is referenced by:  inatsk  9155  grur1  9197
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