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Theorem rankcf 9172
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 8230 . . 3  |-  ( rank `  A )  e.  On
2 onzsl 6680 . . 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 7668 . . . 4  |-  -.  A  ~< 
(/)
5 fveq2 5872 . . . . . 6  |-  ( (
rank `  A )  =  (/)  ->  ( cf `  ( rank `  A
) )  =  ( cf `  (/) ) )
6 cf0 8648 . . . . . 6  |-  ( cf `  (/) )  =  (/)
75, 6syl6eq 2514 . . . . 5  |-  ( (
rank `  A )  =  (/)  ->  ( cf `  ( rank `  A
) )  =  (/) )
87breq2d 4468 . . . 4  |-  ( (
rank `  A )  =  (/)  ->  ( A  ~<  ( cf `  ( rank `  A ) )  <-> 
A  ~<  (/) ) )
94, 8mtbiri 303 . . 3  |-  ( (
rank `  A )  =  (/)  ->  -.  A  ~<  ( cf `  ( rank `  A ) ) )
10 fveq2 5872 . . . . . . 7  |-  ( (
rank `  A )  =  suc  x  ->  ( cf `  ( rank `  A
) )  =  ( cf `  suc  x
) )
11 cfsuc 8654 . . . . . . 7  |-  ( x  e.  On  ->  ( cf `  suc  x )  =  1o )
1210, 11sylan9eqr 2520 . . . . . 6  |-  ( ( x  e.  On  /\  ( rank `  A )  =  suc  x )  -> 
( cf `  ( rank `  A ) )  =  1o )
13 nsuceq0 4967 . . . . . . . . 9  |-  suc  x  =/=  (/)
14 neeq1 2738 . . . . . . . . 9  |-  ( (
rank `  A )  =  suc  x  ->  (
( rank `  A )  =/=  (/)  <->  suc  x  =/=  (/) ) )
1513, 14mpbiri 233 . . . . . . . 8  |-  ( (
rank `  A )  =  suc  x  ->  ( rank `  A )  =/=  (/) )
16 fveq2 5872 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  ( rank `  A )  =  (
rank `  (/) ) )
17 0elon 4940 . . . . . . . . . . . . 13  |-  (/)  e.  On
18 r1fnon 8202 . . . . . . . . . . . . . 14  |-  R1  Fn  On
19 fndm 5686 . . . . . . . . . . . . . 14  |-  ( R1  Fn  On  ->  dom  R1  =  On )
2018, 19ax-mp 5 . . . . . . . . . . . . 13  |-  dom  R1  =  On
2117, 20eleqtrri 2544 . . . . . . . . . . . 12  |-  (/)  e.  dom  R1
22 rankonid 8264 . . . . . . . . . . . 12  |-  ( (/)  e.  dom  R1  <->  ( rank `  (/) )  =  (/) )
2321, 22mpbi 208 . . . . . . . . . . 11  |-  ( rank `  (/) )  =  (/)
2416, 23syl6eq 2514 . . . . . . . . . 10  |-  ( A  =  (/)  ->  ( rank `  A )  =  (/) )
2524necon3i 2697 . . . . . . . . 9  |-  ( (
rank `  A )  =/=  (/)  ->  A  =/=  (/) )
26 rankvaln 8234 . . . . . . . . . . 11  |-  ( -.  A  e.  U. ( R1 " On )  -> 
( rank `  A )  =  (/) )
2726necon1ai 2688 . . . . . . . . . 10  |-  ( (
rank `  A )  =/=  (/)  ->  A  e.  U. ( R1 " On ) )
28 breq2 4460 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( 1o 
~<_  y  <->  1o  ~<_  A )
)
29 neeq1 2738 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
y  =/=  (/)  <->  A  =/=  (/) ) )
30 0sdom1dom 7736 . . . . . . . . . . . 12  |-  ( (/)  ~< 
y  <->  1o  ~<_  y )
31 vex 3112 . . . . . . . . . . . . 13  |-  y  e. 
_V
32310sdom 7667 . . . . . . . . . . . 12  |-  ( (/)  ~< 
y  <->  y  =/=  (/) )
3330, 32bitr3i 251 . . . . . . . . . . 11  |-  ( 1o  ~<_  y  <->  y  =/=  (/) )
3428, 29, 33vtoclbg 3168 . . . . . . . . . 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 4476 . . . . 5  |-  ( ( x  e.  On  /\  ( rank `  A )  =  suc  x )  -> 
( cf `  ( rank `  A ) )  ~<_  A )
4039rexlimiva 2945 . . . 4  |-  ( E. x  e.  On  ( rank `  A )  =  suc  x  ->  ( cf `  ( rank `  A
) )  ~<_  A )
41 domnsym 7662 . . . 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 4945 . . . . . . . . . . . . . . . . 17  |-  -.  Lim  (/)
44 limeq 4899 . . . . . . . . . . . . . . . . 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 8240 . . . . . . . . . . . . . 14  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  U. ( R1
" On ) )
4947, 48syl 16 . . . . . . . . . . . . 13  |-  ( Lim  ( rank `  A
)  ->  A  C_  U. ( R1 " On ) )
5049sselda 3499 . . . . . . . . . . . 12  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  x  e.  U. ( R1 " On ) )
51 ranksnb 8262 . . . . . . . . . . . 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 8259 . . . . . . . . . . . . . 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 6683 . . . . . . . . . . . . 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 2545 . . . . . . . . . 10  |-  ( ( Lim  ( rank `  A
)  /\  x  e.  A )  ->  ( rank `  { x }
)  e.  ( rank `  A ) )
59 eleq1a 2540 . . . . . . . . . 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 2949 . . . . . . . 8  |-  ( Lim  ( rank `  A
)  ->  ( E. x  e.  A  w  =  ( rank `  {
x } )  ->  w  e.  ( rank `  A ) ) )
6261abssdv 3570 . . . . . . 7  |-  ( Lim  ( rank `  A
)  ->  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }  C_  ( rank `  A ) )
63 snex 4697 . . . . . . . . . . . . 13  |-  { x }  e.  _V
6463dfiun2 4366 . . . . . . . . . . . 12  |-  U_ x  e.  A  { x }  =  U. { y  |  E. x  e.  A  y  =  {
x } }
65 iunid 4387 . . . . . . . . . . . 12  |-  U_ x  e.  A  { x }  =  A
6664, 65eqtr3i 2488 . . . . . . . . . . 11  |-  U. {
y  |  E. x  e.  A  y  =  { x } }  =  A
6766fveq2i 5875 . . . . . . . . . 10  |-  ( rank `  U. { y  |  E. x  e.  A  y  =  { x } } )  =  (
rank `  A )
6848sselda 3499 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  U. ( R1 " On )  /\  x  e.  A )  ->  x  e.  U. ( R1 " On ) )
69 snwf 8244 . . . . . . . . . . . . . . 15  |-  ( x  e.  U. ( R1
" On )  ->  { x }  e.  U. ( R1 " On ) )
70 eleq1a 2540 . . . . . . . . . . . . . . 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 2949 . . . . . . . . . . . . 13  |-  ( A  e.  U. ( R1
" On )  -> 
( E. x  e.  A  y  =  {
x }  ->  y  e.  U. ( R1 " On ) ) )
7372abssdv 3570 . . . . . . . . . . . 12  |-  ( A  e.  U. ( R1
" On )  ->  { y  |  E. x  e.  A  y  =  { x } }  C_ 
U. ( R1 " On ) )
74 abrexexg 6774 . . . . . . . . . . . . 13  |-  ( A  e.  U. ( R1
" On )  ->  { y  |  E. x  e.  A  y  =  { x } }  e.  _V )
75 eleq1 2529 . . . . . . . . . . . . . 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 3520 . . . . . . . . . . . . . 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 3112 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
7877r1elss 8241 . . . . . . . . . . . . . 14  |-  ( z  e.  U. ( R1
" On )  <->  z  C_  U. ( R1 " On ) )
7975, 76, 78vtoclbg 3168 . . . . . . . . . . . . 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 8288 . . . . . . . . . . 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 2512 . . . . . . . . 9  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  U_ z  e.  {
y  |  E. x  e.  A  y  =  { x } } 
( rank `  z )
)
85 fvex 5882 . . . . . . . . . . 11  |-  ( rank `  z )  e.  _V
8685dfiun2 4366 . . . . . . . . . 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 5872 . . . . . . . . . . . 12  |-  ( z  =  { x }  ->  ( rank `  z
)  =  ( rank `  { x } ) )
8863, 87abrexco 6157 . . . . . . . . . . 11  |-  { w  |  E. z  e.  {
y  |  E. x  e.  A  y  =  { x } }
w  =  ( rank `  z ) }  =  { w  |  E. x  e.  A  w  =  ( rank `  {
x } ) }
8988unieqi 4260 . . . . . . . . . 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 2486 . . . . . . . . 9  |-  U_ z  e.  { y  |  E. x  e.  A  y  =  { x } } 
( rank `  z )  =  U. { w  |  E. x  e.  A  w  =  ( rank `  { x } ) }
9184, 90syl6req 2515 . . . . . . . 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 5882 . . . . . . . 8  |-  ( rank `  A )  e.  _V
9493cfslb 8663 . . . . . . 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 5872 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( rank `  y )  =  ( rank `  A
) )
9796fveq2d 5876 . . . . . . . . . 10  |-  ( y  =  A  ->  ( cf `  ( rank `  y
) )  =  ( cf `  ( rank `  A ) ) )
98 breq12 4461 . . . . . . . . . 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 3055 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( E. x  e.  y  w  =  ( rank `  { x } )  <->  E. x  e.  A  w  =  ( rank `  { x } ) ) )
101100abbidv 2593 . . . . . . . . . 10  |-  ( y  =  A  ->  { w  |  E. x  e.  y  w  =  ( rank `  { x } ) }  =  { w  |  E. x  e.  A  w  =  ( rank `  { x } ) } )
102 breq12 4461 . . . . . . . . . 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 2457 . . . . . . . . . 10  |-  ( x  e.  y  |->  ( rank `  { x } ) )  =  ( x  e.  y  |->  ( rank `  { x } ) )
106105rnmpt 5258 . . . . . . . . 9  |-  ran  (
x  e.  y  |->  (
rank `  { x } ) )  =  { w  |  E. x  e.  y  w  =  ( rank `  {
x } ) }
107 cfon 8652 . . . . . . . . . . 11  |-  ( cf `  ( rank `  y
) )  e.  On
108 sdomdom 7562 . . . . . . . . . . 11  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  y  ~<_  ( cf `  ( rank `  y
) ) )
109 ondomen 8435 . . . . . . . . . . 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 5882 . . . . . . . . . . . 12  |-  ( rank `  { x } )  e.  _V
112111, 105fnmpti 5715 . . . . . . . . . . 11  |-  ( x  e.  y  |->  ( rank `  { x } ) )  Fn  y
113 dffn4 5807 . . . . . . . . . . 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 8455 . . . . . . . . . 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 4490 . . . . . . . 8  |-  ( y 
~<  ( cf `  ( rank `  y ) )  ->  { w  |  E. x  e.  y  w  =  ( rank `  { x } ) }  ~<_  y )
118104, 117vtoclg 3167 . . . . . . 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 7587 . . . . . . 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 1395    e. wcel 1819   {cab 2442    =/= wne 2652   E.wrex 2808   _Vcvv 3109    C_ wss 3471   (/)c0 3793   {csn 4032   U.cuni 4251   U_ciun 4332   class class class wbr 4456    |-> cmpt 4515   Oncon0 4887   Lim wlim 4888   suc csuc 4889   dom cdm 5008   ran crn 5009   "cima 5011    Fn wfn 5589   -onto->wfo 5592   ` cfv 5594   1oc1o 7141    ~<_ cdom 7533    ~< csdm 7534   R1cr1 8197   rankcrnk 8198   cardccrd 8333   cfccf 8335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-r1 8199  df-rank 8200  df-card 8337  df-cf 8339  df-acn 8340
This theorem is referenced by:  inatsk  9173  grur1  9215
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