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Theorem cfss 8430
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 8427 . . . . 5  |-  ( Lim 
A  ->  ( cf `  A )  =  |^|_ x  e.  { x  e. 
~P A  |  U. x  =  A } 
( card `  x )
)
3 fvex 5698 . . . . . . 7  |-  ( card `  x )  e.  _V
43dfiin2 4202 . . . . . 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 8110 . . . . . . . . . 10  |-  ( card `  x )  e.  On
6 eleq1 2501 . . . . . . . . . 10  |-  ( y  =  ( card `  x
)  ->  ( y  e.  On  <->  ( card `  x
)  e.  On ) )
75, 6mpbiri 233 . . . . . . . . 9  |-  ( y  =  ( card `  x
)  ->  y  e.  On )
87rexlimivw 2835 . . . . . . . 8  |-  ( E. x  e.  { x  e.  ~P A  |  U. x  =  A }
y  =  ( card `  x )  ->  y  e.  On )
98abssi 3424 . . . . . . 7  |-  { y  |  E. x  e. 
{ x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) }  C_  On
10 limuni 4775 . . . . . . . . . . . 12  |-  ( Lim 
A  ->  A  =  U. A )
1110eqcomd 2446 . . . . . . . . . . 11  |-  ( Lim 
A  ->  U. A  =  A )
12 fveq2 5688 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  ( card `  x )  =  ( card `  A
) )
1312eqcomd 2446 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  ( card `  A )  =  ( card `  x
) )
1413biantrud 504 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  ( U. A  =  A  <->  ( U. A  =  A  /\  ( card `  A
)  =  ( card `  x ) ) ) )
15 unieq 4096 . . . . . . . . . . . . . . . 16  |-  ( x  =  A  ->  U. x  =  U. A )
1615eqeq1d 2449 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  ( U. x  =  A  <->  U. A  =  A ) )
171pwid 3871 . . . . . . . . . . . . . . . . 17  |-  A  e. 
~P A
18 eleq1 2501 . . . . . . . . . . . . . . . . 17  |-  ( x  =  A  ->  (
x  e.  ~P A  <->  A  e.  ~P A ) )
1917, 18mpbiri 233 . . . . . . . . . . . . . . . 16  |-  ( x  =  A  ->  x  e.  ~P A )
2019biantrurd 505 . . . . . . . . . . . . . . 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 699 . . . . . . . . . . . . 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 3061 . . . . . . . . . . 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 2719 . . . . . . . . . . 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 2895 . . . . . . . . . . . . 13  |-  ( x  e.  { x  e. 
~P A  |  U. x  =  A }  <->  ( x  e.  ~P A  /\  U. x  =  A ) )
2827anbi1i 690 . . . . . . . . . . . 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 1639 . . . . . . . . . . 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 5698 . . . . . . . . . 10  |-  ( card `  A )  e.  _V
33 eqeq1 2447 . . . . . . . . . . 11  |-  ( y  =  ( card `  A
)  ->  ( y  =  ( card `  x
)  <->  ( card `  A
)  =  ( card `  x ) ) )
3433rexbidv 2734 . . . . . . . . . 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 3061 . . . . . . . . 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 3653 . . . . . . . 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 6405 . . . . . . 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 658 . . . . . 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 2525 . . . . 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 2515 . . . 4  |-  ( Lim 
A  ->  ( cf `  A )  e.  {
y  |  E. x  e.  { x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) } )
43 fvex 5698 . . . . 5  |-  ( cf `  A )  e.  _V
44 eqeq1 2447 . . . . . 6  |-  ( y  =  ( cf `  A
)  ->  ( y  =  ( card `  x
)  <->  ( cf `  A
)  =  ( card `  x ) ) )
4544rexbidv 2734 . . . . 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 3103 . . . 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 2719 . . 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 750 . . . . . . . 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 456 . . . . . 6  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  e.  ~P A )
5352elpwid 3867 . . . . 5  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  C_  A
)
54 simpl 454 . . . . . . 7  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  Lim  A )
55 vex 2973 . . . . . . . . . 10  |-  x  e. 
_V
56 limord 4774 . . . . . . . . . . . 12  |-  ( Lim 
A  ->  Ord  A )
57 ordsson 6400 . . . . . . . . . . . 12  |-  ( Ord 
A  ->  A  C_  On )
5856, 57syl 16 . . . . . . . . . . 11  |-  ( Lim 
A  ->  A  C_  On )
59 sstr 3361 . . . . . . . . . . 11  |-  ( ( x  C_  A  /\  A  C_  On )  ->  x  C_  On )
6058, 59sylan2 471 . . . . . . . . . 10  |-  ( ( x  C_  A  /\  Lim  A )  ->  x  C_  On )
61 onssnum 8206 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  x  C_  On )  ->  x  e.  dom  card )
6255, 60, 61sylancr 658 . . . . . . . . 9  |-  ( ( x  C_  A  /\  Lim  A )  ->  x  e.  dom  card )
63 cardid2 8119 . . . . . . . . 9  |-  ( x  e.  dom  card  ->  (
card `  x )  ~~  x )
6462, 63syl 16 . . . . . . . 8  |-  ( ( x  C_  A  /\  Lim  A )  ->  ( card `  x )  ~~  x )
6564ensymd 7356 . . . . . . 7  |-  ( ( x  C_  A  /\  Lim  A )  ->  x  ~~  ( card `  x
) )
6653, 54, 65syl2anc 656 . . . . . 6  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  ~~  ( card `  x ) )
67 simprr 751 . . . . . 6  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  ( cf `  A
)  =  ( card `  x ) )
6866, 67breqtrrd 4315 . . . . 5  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  ~~  ( cf `  A ) )
6951simprd 460 . . . . 5  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  U. x  =  A )
7053, 68, 693jca 1163 . . . 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 1681 . 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 960    = wceq 1364   E.wex 1591    e. wcel 1761   {cab 2427    =/= wne 2604   E.wrex 2714   {crab 2717   _Vcvv 2970    C_ wss 3325   (/)c0 3634   ~Pcpw 3857   U.cuni 4088   |^|cint 4125   |^|_ciin 4169   class class class wbr 4289   Ord word 4714   Oncon0 4715   Lim wlim 4716   dom cdm 4836   ` cfv 5415    ~~ cen 7303   cardccrd 8101   cfccf 8103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-recs 6828  df-er 7097  df-en 7307  df-dom 7308  df-card 8105  df-cf 8107
This theorem is referenced by: (None)
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