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Theorem cfss 8713
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 8710 . . . . 5  |-  ( Lim 
A  ->  ( cf `  A )  =  |^|_ x  e.  { x  e. 
~P A  |  U. x  =  A } 
( card `  x )
)
3 fvex 5889 . . . . . . 7  |-  ( card `  x )  e.  _V
43dfiin2 4304 . . . . . 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 8396 . . . . . . . . . 10  |-  ( card `  x )  e.  On
6 eleq1 2537 . . . . . . . . . 10  |-  ( y  =  ( card `  x
)  ->  ( y  e.  On  <->  ( card `  x
)  e.  On ) )
75, 6mpbiri 241 . . . . . . . . 9  |-  ( y  =  ( card `  x
)  ->  y  e.  On )
87rexlimivw 2869 . . . . . . . 8  |-  ( E. x  e.  { x  e.  ~P A  |  U. x  =  A }
y  =  ( card `  x )  ->  y  e.  On )
98abssi 3490 . . . . . . 7  |-  { y  |  E. x  e. 
{ x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) }  C_  On
10 limuni 5490 . . . . . . . . . . . 12  |-  ( Lim 
A  ->  A  =  U. A )
1110eqcomd 2477 . . . . . . . . . . 11  |-  ( Lim 
A  ->  U. A  =  A )
12 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  ( card `  x )  =  ( card `  A
) )
1312eqcomd 2477 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  ( card `  A )  =  ( card `  x
) )
1413biantrud 515 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  ( U. A  =  A  <->  ( U. A  =  A  /\  ( card `  A
)  =  ( card `  x ) ) ) )
15 unieq 4198 . . . . . . . . . . . . . . . 16  |-  ( x  =  A  ->  U. x  =  U. A )
1615eqeq1d 2473 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  ( U. x  =  A  <->  U. A  =  A ) )
171pwid 3956 . . . . . . . . . . . . . . . . 17  |-  A  e. 
~P A
18 eleq1 2537 . . . . . . . . . . . . . . . . 17  |-  ( x  =  A  ->  (
x  e.  ~P A  <->  A  e.  ~P A ) )
1917, 18mpbiri 241 . . . . . . . . . . . . . . . 16  |-  ( x  =  A  ->  x  e.  ~P A )
2019biantrurd 516 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  ( U. x  =  A  <->  ( x  e.  ~P A  /\  U. x  =  A ) ) )
2116, 20bitr3d 263 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  ( U. A  =  A  <->  ( x  e.  ~P A  /\  U. x  =  A ) ) )
2221anbi1d 719 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  (
( U. A  =  A  /\  ( card `  A )  =  (
card `  x )
)  <->  ( ( x  e.  ~P A  /\  U. x  =  A )  /\  ( card `  A
)  =  ( card `  x ) ) ) )
2314, 22bitr2d 262 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
( ( x  e. 
~P A  /\  U. x  =  A )  /\  ( card `  A
)  =  ( card `  x ) )  <->  U. A  =  A ) )
241, 23spcev 3127 . . . . . . . . . . 11  |-  ( U. A  =  A  ->  E. x ( ( x  e.  ~P A  /\  U. x  =  A )  /\  ( card `  A
)  =  ( card `  x ) ) )
2511, 24syl 17 . . . . . . . . . 10  |-  ( Lim 
A  ->  E. x
( ( x  e. 
~P A  /\  U. x  =  A )  /\  ( card `  A
)  =  ( card `  x ) ) )
26 df-rex 2762 . . . . . . . . . . 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 2953 . . . . . . . . . . . . 13  |-  ( x  e.  { x  e. 
~P A  |  U. x  =  A }  <->  ( x  e.  ~P A  /\  U. x  =  A ) )
2827anbi1i 709 . . . . . . . . . . . 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 1726 . . . . . . . . . . 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 257 . . . . . . . . . 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 217 . . . . . . . . 9  |-  ( Lim 
A  ->  E. x  e.  { x  e.  ~P A  |  U. x  =  A }  ( card `  A )  =  (
card `  x )
)
32 fvex 5889 . . . . . . . . . 10  |-  ( card `  A )  e.  _V
33 eqeq1 2475 . . . . . . . . . . 11  |-  ( y  =  ( card `  A
)  ->  ( y  =  ( card `  x
)  <->  ( card `  A
)  =  ( card `  x ) ) )
3433rexbidv 2892 . . . . . . . . . 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 3127 . . . . . . . . 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 17 . . . . . . . 8  |-  ( Lim 
A  ->  E. y E. x  e.  { x  e.  ~P A  |  U. x  =  A }
y  =  ( card `  x ) )
37 abn0 3754 . . . . . . . 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 217 . . . . . . 7  |-  ( Lim 
A  ->  { y  |  E. x  e.  {
x  e.  ~P A  |  U. x  =  A } y  =  (
card `  x ) }  =/=  (/) )
39 onint 6641 . . . . . . 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 676 . . . . . 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 2553 . . . . 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 2549 . . . 4  |-  ( Lim 
A  ->  ( cf `  A )  e.  {
y  |  E. x  e.  { x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) } )
43 fvex 5889 . . . . 5  |-  ( cf `  A )  e.  _V
44 eqeq1 2475 . . . . . 6  |-  ( y  =  ( cf `  A
)  ->  ( y  =  ( card `  x
)  <->  ( cf `  A
)  =  ( card `  x ) ) )
4544rexbidv 2892 . . . . 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 3173 . . . 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 201 . . 3  |-  ( Lim 
A  ->  E. x  e.  { x  e.  ~P A  |  U. x  =  A }  ( cf `  A )  =  (
card `  x )
)
48 df-rex 2762 . . 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 201 . 2  |-  ( Lim 
A  ->  E. x
( x  e.  {
x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A )  =  (
card `  x )
) )
50 simprl 772 . . . . . . . 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 201 . . . . . . 7  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  ( x  e. 
~P A  /\  U. x  =  A )
)
5251simpld 466 . . . . . 6  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  e.  ~P A )
5352elpwid 3952 . . . . 5  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  C_  A
)
54 simpl 464 . . . . . . 7  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  Lim  A )
55 vex 3034 . . . . . . . . . 10  |-  x  e. 
_V
56 limord 5489 . . . . . . . . . . . 12  |-  ( Lim 
A  ->  Ord  A )
57 ordsson 6635 . . . . . . . . . . . 12  |-  ( Ord 
A  ->  A  C_  On )
5856, 57syl 17 . . . . . . . . . . 11  |-  ( Lim 
A  ->  A  C_  On )
59 sstr 3426 . . . . . . . . . . 11  |-  ( ( x  C_  A  /\  A  C_  On )  ->  x  C_  On )
6058, 59sylan2 482 . . . . . . . . . 10  |-  ( ( x  C_  A  /\  Lim  A )  ->  x  C_  On )
61 onssnum 8489 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  x  C_  On )  ->  x  e.  dom  card )
6255, 60, 61sylancr 676 . . . . . . . . 9  |-  ( ( x  C_  A  /\  Lim  A )  ->  x  e.  dom  card )
63 cardid2 8405 . . . . . . . . 9  |-  ( x  e.  dom  card  ->  (
card `  x )  ~~  x )
6462, 63syl 17 . . . . . . . 8  |-  ( ( x  C_  A  /\  Lim  A )  ->  ( card `  x )  ~~  x )
6564ensymd 7638 . . . . . . 7  |-  ( ( x  C_  A  /\  Lim  A )  ->  x  ~~  ( card `  x
) )
6653, 54, 65syl2anc 673 . . . . . 6  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  ~~  ( card `  x ) )
67 simprr 774 . . . . . 6  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  ( cf `  A
)  =  ( card `  x ) )
6866, 67breqtrrd 4422 . . . . 5  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  ~~  ( cf `  A ) )
6951simprd 470 . . . . 5  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  U. x  =  A )
7053, 68, 693jca 1210 . . . 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 441 . . 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 1772 . 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 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457    =/= wne 2641   E.wrex 2757   {crab 2760   _Vcvv 3031    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   U.cuni 4190   |^|cint 4226   |^|_ciin 4270   class class class wbr 4395   dom cdm 4839   Ord word 5429   Oncon0 5430   Lim wlim 5431   ` cfv 5589    ~~ cen 7584   cardccrd 8387   cfccf 8389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-wrecs 7046  df-recs 7108  df-er 7381  df-en 7588  df-dom 7589  df-card 8391  df-cf 8393
This theorem is referenced by: (None)
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