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Theorem cfss 8646
Description: There is a cofinal subset of  A of cardinality  ( cf `  A ). (Contributed by Mario Carneiro, 24-Jun-2013.)
Hypothesis
Ref Expression
cfss.1  |-  A  e. 
_V
Assertion
Ref Expression
cfss  |-  ( Lim 
A  ->  E. x
( x  C_  A  /\  x  ~~  ( cf `  A )  /\  U. x  =  A )
)
Distinct variable group:    x, A

Proof of Theorem cfss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cfss.1 . . . . . 6  |-  A  e. 
_V
21cflim3 8643 . . . . 5  |-  ( Lim 
A  ->  ( cf `  A )  =  |^|_ x  e.  { x  e. 
~P A  |  U. x  =  A } 
( card `  x )
)
3 fvex 5876 . . . . . . 7  |-  ( card `  x )  e.  _V
43dfiin2 4360 . . . . . 6  |-  |^|_ x  e.  { x  e.  ~P A  |  U. x  =  A }  ( card `  x )  =  |^| { y  |  E. x  e.  { x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) }
5 cardon 8326 . . . . . . . . . 10  |-  ( card `  x )  e.  On
6 eleq1 2539 . . . . . . . . . 10  |-  ( y  =  ( card `  x
)  ->  ( y  e.  On  <->  ( card `  x
)  e.  On ) )
75, 6mpbiri 233 . . . . . . . . 9  |-  ( y  =  ( card `  x
)  ->  y  e.  On )
87rexlimivw 2952 . . . . . . . 8  |-  ( E. x  e.  { x  e.  ~P A  |  U. x  =  A }
y  =  ( card `  x )  ->  y  e.  On )
98abssi 3575 . . . . . . 7  |-  { y  |  E. x  e. 
{ x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) }  C_  On
10 limuni 4938 . . . . . . . . . . . 12  |-  ( Lim 
A  ->  A  =  U. A )
1110eqcomd 2475 . . . . . . . . . . 11  |-  ( Lim 
A  ->  U. A  =  A )
12 fveq2 5866 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  ( card `  x )  =  ( card `  A
) )
1312eqcomd 2475 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  ( card `  A )  =  ( card `  x
) )
1413biantrud 507 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  ( U. A  =  A  <->  ( U. A  =  A  /\  ( card `  A
)  =  ( card `  x ) ) ) )
15 unieq 4253 . . . . . . . . . . . . . . . 16  |-  ( x  =  A  ->  U. x  =  U. A )
1615eqeq1d 2469 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  ( U. x  =  A  <->  U. A  =  A ) )
171pwid 4024 . . . . . . . . . . . . . . . . 17  |-  A  e. 
~P A
18 eleq1 2539 . . . . . . . . . . . . . . . . 17  |-  ( x  =  A  ->  (
x  e.  ~P A  <->  A  e.  ~P A ) )
1917, 18mpbiri 233 . . . . . . . . . . . . . . . 16  |-  ( x  =  A  ->  x  e.  ~P A )
2019biantrurd 508 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  ( U. x  =  A  <->  ( x  e.  ~P A  /\  U. x  =  A ) ) )
2116, 20bitr3d 255 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  ( U. A  =  A  <->  ( x  e.  ~P A  /\  U. x  =  A ) ) )
2221anbi1d 704 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  (
( U. A  =  A  /\  ( card `  A )  =  (
card `  x )
)  <->  ( ( x  e.  ~P A  /\  U. x  =  A )  /\  ( card `  A
)  =  ( card `  x ) ) ) )
2314, 22bitr2d 254 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
( ( x  e. 
~P A  /\  U. x  =  A )  /\  ( card `  A
)  =  ( card `  x ) )  <->  U. A  =  A ) )
241, 23spcev 3205 . . . . . . . . . . 11  |-  ( U. A  =  A  ->  E. x ( ( x  e.  ~P A  /\  U. x  =  A )  /\  ( card `  A
)  =  ( card `  x ) ) )
2511, 24syl 16 . . . . . . . . . 10  |-  ( Lim 
A  ->  E. x
( ( x  e. 
~P A  /\  U. x  =  A )  /\  ( card `  A
)  =  ( card `  x ) ) )
26 df-rex 2820 . . . . . . . . . . 11  |-  ( E. x  e.  { x  e.  ~P A  |  U. x  =  A } 
( card `  A )  =  ( card `  x
)  <->  E. x ( x  e.  { x  e. 
~P A  |  U. x  =  A }  /\  ( card `  A
)  =  ( card `  x ) ) )
27 rabid 3038 . . . . . . . . . . . . 13  |-  ( x  e.  { x  e. 
~P A  |  U. x  =  A }  <->  ( x  e.  ~P A  /\  U. x  =  A ) )
2827anbi1i 695 . . . . . . . . . . . 12  |-  ( ( x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( card `  A
)  =  ( card `  x ) )  <->  ( (
x  e.  ~P A  /\  U. x  =  A )  /\  ( card `  A )  =  (
card `  x )
) )
2928exbii 1644 . . . . . . . . . . 11  |-  ( E. x ( x  e. 
{ x  e.  ~P A  |  U. x  =  A }  /\  ( card `  A )  =  ( card `  x
) )  <->  E. x
( ( x  e. 
~P A  /\  U. x  =  A )  /\  ( card `  A
)  =  ( card `  x ) ) )
3026, 29bitri 249 . . . . . . . . . 10  |-  ( E. x  e.  { x  e.  ~P A  |  U. x  =  A } 
( card `  A )  =  ( card `  x
)  <->  E. x ( ( x  e.  ~P A  /\  U. x  =  A )  /\  ( card `  A )  =  (
card `  x )
) )
3125, 30sylibr 212 . . . . . . . . 9  |-  ( Lim 
A  ->  E. x  e.  { x  e.  ~P A  |  U. x  =  A }  ( card `  A )  =  (
card `  x )
)
32 fvex 5876 . . . . . . . . . 10  |-  ( card `  A )  e.  _V
33 eqeq1 2471 . . . . . . . . . . 11  |-  ( y  =  ( card `  A
)  ->  ( y  =  ( card `  x
)  <->  ( card `  A
)  =  ( card `  x ) ) )
3433rexbidv 2973 . . . . . . . . . 10  |-  ( y  =  ( card `  A
)  ->  ( E. x  e.  { x  e.  ~P A  |  U. x  =  A }
y  =  ( card `  x )  <->  E. x  e.  { x  e.  ~P A  |  U. x  =  A }  ( card `  A )  =  (
card `  x )
) )
3532, 34spcev 3205 . . . . . . . . 9  |-  ( E. x  e.  { x  e.  ~P A  |  U. x  =  A } 
( card `  A )  =  ( card `  x
)  ->  E. y E. x  e.  { x  e.  ~P A  |  U. x  =  A }
y  =  ( card `  x ) )
3631, 35syl 16 . . . . . . . 8  |-  ( Lim 
A  ->  E. y E. x  e.  { x  e.  ~P A  |  U. x  =  A }
y  =  ( card `  x ) )
37 abn0 3804 . . . . . . . 8  |-  ( { y  |  E. x  e.  { x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) }  =/=  (/)  <->  E. y E. x  e.  { x  e.  ~P A  |  U. x  =  A }
y  =  ( card `  x ) )
3836, 37sylibr 212 . . . . . . 7  |-  ( Lim 
A  ->  { y  |  E. x  e.  {
x  e.  ~P A  |  U. x  =  A } y  =  (
card `  x ) }  =/=  (/) )
39 onint 6615 . . . . . . 7  |-  ( ( { y  |  E. x  e.  { x  e.  ~P A  |  U. x  =  A }
y  =  ( card `  x ) }  C_  On  /\  { y  |  E. x  e.  {
x  e.  ~P A  |  U. x  =  A } y  =  (
card `  x ) }  =/=  (/) )  ->  |^| { y  |  E. x  e. 
{ x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) }  e.  {
y  |  E. x  e.  { x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) } )
409, 38, 39sylancr 663 . . . . . 6  |-  ( Lim 
A  ->  |^| { y  |  E. x  e. 
{ x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) }  e.  {
y  |  E. x  e.  { x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) } )
414, 40syl5eqel 2559 . . . . 5  |-  ( Lim 
A  ->  |^|_ x  e. 
{ x  e.  ~P A  |  U. x  =  A }  ( card `  x )  e.  {
y  |  E. x  e.  { x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) } )
422, 41eqeltrd 2555 . . . 4  |-  ( Lim 
A  ->  ( cf `  A )  e.  {
y  |  E. x  e.  { x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) } )
43 fvex 5876 . . . . 5  |-  ( cf `  A )  e.  _V
44 eqeq1 2471 . . . . . 6  |-  ( y  =  ( cf `  A
)  ->  ( y  =  ( card `  x
)  <->  ( cf `  A
)  =  ( card `  x ) ) )
4544rexbidv 2973 . . . . 5  |-  ( y  =  ( cf `  A
)  ->  ( E. x  e.  { x  e.  ~P A  |  U. x  =  A }
y  =  ( card `  x )  <->  E. x  e.  { x  e.  ~P A  |  U. x  =  A }  ( cf `  A )  =  (
card `  x )
) )
4643, 45elab 3250 . . . 4  |-  ( ( cf `  A )  e.  { y  |  E. x  e.  {
x  e.  ~P A  |  U. x  =  A } y  =  (
card `  x ) } 
<->  E. x  e.  {
x  e.  ~P A  |  U. x  =  A }  ( cf `  A
)  =  ( card `  x ) )
4742, 46sylib 196 . . 3  |-  ( Lim 
A  ->  E. x  e.  { x  e.  ~P A  |  U. x  =  A }  ( cf `  A )  =  (
card `  x )
)
48 df-rex 2820 . . 3  |-  ( E. x  e.  { x  e.  ~P A  |  U. x  =  A } 
( cf `  A
)  =  ( card `  x )  <->  E. x
( x  e.  {
x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A )  =  (
card `  x )
) )
4947, 48sylib 196 . 2  |-  ( Lim 
A  ->  E. x
( x  e.  {
x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A )  =  (
card `  x )
) )
50 simprl 755 . . . . . . . 8  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  e.  {
x  e.  ~P A  |  U. x  =  A } )
5150, 27sylib 196 . . . . . . 7  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  ( x  e. 
~P A  /\  U. x  =  A )
)
5251simpld 459 . . . . . 6  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  e.  ~P A )
5352elpwid 4020 . . . . 5  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  C_  A
)
54 simpl 457 . . . . . . 7  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  Lim  A )
55 vex 3116 . . . . . . . . . 10  |-  x  e. 
_V
56 limord 4937 . . . . . . . . . . . 12  |-  ( Lim 
A  ->  Ord  A )
57 ordsson 6610 . . . . . . . . . . . 12  |-  ( Ord 
A  ->  A  C_  On )
5856, 57syl 16 . . . . . . . . . . 11  |-  ( Lim 
A  ->  A  C_  On )
59 sstr 3512 . . . . . . . . . . 11  |-  ( ( x  C_  A  /\  A  C_  On )  ->  x  C_  On )
6058, 59sylan2 474 . . . . . . . . . 10  |-  ( ( x  C_  A  /\  Lim  A )  ->  x  C_  On )
61 onssnum 8422 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  x  C_  On )  ->  x  e.  dom  card )
6255, 60, 61sylancr 663 . . . . . . . . 9  |-  ( ( x  C_  A  /\  Lim  A )  ->  x  e.  dom  card )
63 cardid2 8335 . . . . . . . . 9  |-  ( x  e.  dom  card  ->  (
card `  x )  ~~  x )
6462, 63syl 16 . . . . . . . 8  |-  ( ( x  C_  A  /\  Lim  A )  ->  ( card `  x )  ~~  x )
6564ensymd 7567 . . . . . . 7  |-  ( ( x  C_  A  /\  Lim  A )  ->  x  ~~  ( card `  x
) )
6653, 54, 65syl2anc 661 . . . . . 6  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  ~~  ( card `  x ) )
67 simprr 756 . . . . . 6  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  ( cf `  A
)  =  ( card `  x ) )
6866, 67breqtrrd 4473 . . . . 5  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  ~~  ( cf `  A ) )
6951simprd 463 . . . . 5  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  U. x  =  A )
7053, 68, 693jca 1176 . . . 4  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  ( x  C_  A  /\  x  ~~  ( cf `  A )  /\  U. x  =  A ) )
7170ex 434 . . 3  |-  ( Lim 
A  ->  ( (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) )  -> 
( x  C_  A  /\  x  ~~  ( cf `  A )  /\  U. x  =  A )
) )
7271eximdv 1686 . 2  |-  ( Lim 
A  ->  ( E. x ( x  e. 
{ x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A )  =  ( card `  x
) )  ->  E. x
( x  C_  A  /\  x  ~~  ( cf `  A )  /\  U. x  =  A )
) )
7349, 72mpd 15 1  |-  ( Lim 
A  ->  E. x
( x  C_  A  /\  x  ~~  ( cf `  A )  /\  U. x  =  A )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452    =/= wne 2662   E.wrex 2815   {crab 2818   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   U.cuni 4245   |^|cint 4282   |^|_ciin 4326   class class class wbr 4447   Ord word 4877   Oncon0 4878   Lim wlim 4879   dom cdm 4999   ` cfv 5588    ~~ cen 7514   cardccrd 8317   cfccf 8319
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 6577
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-recs 7043  df-er 7312  df-en 7518  df-dom 7519  df-card 8321  df-cf 8323
This theorem is referenced by: (None)
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