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Theorem cfss 7907
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 7904 . . . . 5  |-  ( Lim 
A  ->  ( cf `  A )  =  |^|_ x  e.  { x  e. 
~P A  |  U. x  =  A } 
( card `  x )
)
3 fvex 5555 . . . . . . 7  |-  ( card `  x )  e.  _V
43dfiin2 3954 . . . . . 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 7593 . . . . . . . . . 10  |-  ( card `  x )  e.  On
6 eleq1 2356 . . . . . . . . . 10  |-  ( y  =  ( card `  x
)  ->  ( y  e.  On  <->  ( card `  x
)  e.  On ) )
75, 6mpbiri 224 . . . . . . . . 9  |-  ( y  =  ( card `  x
)  ->  y  e.  On )
87rexlimivw 2676 . . . . . . . 8  |-  ( E. x  e.  { x  e.  ~P A  |  U. x  =  A }
y  =  ( card `  x )  ->  y  e.  On )
98abssi 3261 . . . . . . 7  |-  { y  |  E. x  e. 
{ x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) }  C_  On
10 limuni 4468 . . . . . . . . . . . 12  |-  ( Lim 
A  ->  A  =  U. A )
1110eqcomd 2301 . . . . . . . . . . 11  |-  ( Lim 
A  ->  U. A  =  A )
12 fveq2 5541 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  ( card `  x )  =  ( card `  A
) )
1312eqcomd 2301 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  ( card `  A )  =  ( card `  x
) )
1413biantrud 493 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  ( U. A  =  A  <->  ( U. A  =  A  /\  ( card `  A
)  =  ( card `  x ) ) ) )
15 unieq 3852 . . . . . . . . . . . . . . . 16  |-  ( x  =  A  ->  U. x  =  U. A )
1615eqeq1d 2304 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  ( U. x  =  A  <->  U. A  =  A ) )
171pwid 3651 . . . . . . . . . . . . . . . . 17  |-  A  e. 
~P A
18 eleq1 2356 . . . . . . . . . . . . . . . . 17  |-  ( x  =  A  ->  (
x  e.  ~P A  <->  A  e.  ~P A ) )
1917, 18mpbiri 224 . . . . . . . . . . . . . . . 16  |-  ( x  =  A  ->  x  e.  ~P A )
2019biantrurd 494 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  ( U. x  =  A  <->  ( x  e.  ~P A  /\  U. x  =  A ) ) )
2116, 20bitr3d 246 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  ( U. A  =  A  <->  ( x  e.  ~P A  /\  U. x  =  A ) ) )
2221anbi1d 685 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  (
( U. A  =  A  /\  ( card `  A )  =  (
card `  x )
)  <->  ( ( x  e.  ~P A  /\  U. x  =  A )  /\  ( card `  A
)  =  ( card `  x ) ) ) )
2314, 22bitr2d 245 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
( ( x  e. 
~P A  /\  U. x  =  A )  /\  ( card `  A
)  =  ( card `  x ) )  <->  U. A  =  A ) )
241, 23spcev 2888 . . . . . . . . . . 11  |-  ( U. A  =  A  ->  E. x ( ( x  e.  ~P A  /\  U. x  =  A )  /\  ( card `  A
)  =  ( card `  x ) ) )
2511, 24syl 15 . . . . . . . . . 10  |-  ( Lim 
A  ->  E. x
( ( x  e. 
~P A  /\  U. x  =  A )  /\  ( card `  A
)  =  ( card `  x ) ) )
26 df-rex 2562 . . . . . . . . . . 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 2729 . . . . . . . . . . . . 13  |-  ( x  e.  { x  e. 
~P A  |  U. x  =  A }  <->  ( x  e.  ~P A  /\  U. x  =  A ) )
2827anbi1i 676 . . . . . . . . . . . 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 1572 . . . . . . . . . . 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 240 . . . . . . . . . 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 203 . . . . . . . . 9  |-  ( Lim 
A  ->  E. x  e.  { x  e.  ~P A  |  U. x  =  A }  ( card `  A )  =  (
card `  x )
)
32 fvex 5555 . . . . . . . . . 10  |-  ( card `  A )  e.  _V
33 eqeq1 2302 . . . . . . . . . . 11  |-  ( y  =  ( card `  A
)  ->  ( y  =  ( card `  x
)  <->  ( card `  A
)  =  ( card `  x ) ) )
3433rexbidv 2577 . . . . . . . . . 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 2888 . . . . . . . . 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 15 . . . . . . . 8  |-  ( Lim 
A  ->  E. y E. x  e.  { x  e.  ~P A  |  U. x  =  A }
y  =  ( card `  x ) )
37 abn0 3486 . . . . . . . 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 203 . . . . . . 7  |-  ( Lim 
A  ->  { y  |  E. x  e.  {
x  e.  ~P A  |  U. x  =  A } y  =  (
card `  x ) }  =/=  (/) )
39 onint 4602 . . . . . . 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 644 . . . . . 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 2380 . . . . 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 2370 . . . 4  |-  ( Lim 
A  ->  ( cf `  A )  e.  {
y  |  E. x  e.  { x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) } )
43 fvex 5555 . . . . 5  |-  ( cf `  A )  e.  _V
44 eqeq1 2302 . . . . . 6  |-  ( y  =  ( cf `  A
)  ->  ( y  =  ( card `  x
)  <->  ( cf `  A
)  =  ( card `  x ) ) )
4544rexbidv 2577 . . . . 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 2927 . . . 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 188 . . 3  |-  ( Lim 
A  ->  E. x  e.  { x  e.  ~P A  |  U. x  =  A }  ( cf `  A )  =  (
card `  x )
)
48 df-rex 2562 . . 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 188 . 2  |-  ( Lim 
A  ->  E. x
( x  e.  {
x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A )  =  (
card `  x )
) )
50 simprl 732 . . . . . . . 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 188 . . . . . . 7  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  ( x  e. 
~P A  /\  U. x  =  A )
)
5251simpld 445 . . . . . 6  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  e.  ~P A )
53 elpwi 3646 . . . . . 6  |-  ( x  e.  ~P A  ->  x  C_  A )
5452, 53syl 15 . . . . 5  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  C_  A
)
55 simpl 443 . . . . . . 7  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  Lim  A )
56 vex 2804 . . . . . . . . . 10  |-  x  e. 
_V
57 limord 4467 . . . . . . . . . . . 12  |-  ( Lim 
A  ->  Ord  A )
58 ordsson 4597 . . . . . . . . . . . 12  |-  ( Ord 
A  ->  A  C_  On )
5957, 58syl 15 . . . . . . . . . . 11  |-  ( Lim 
A  ->  A  C_  On )
60 sstr 3200 . . . . . . . . . . 11  |-  ( ( x  C_  A  /\  A  C_  On )  ->  x  C_  On )
6159, 60sylan2 460 . . . . . . . . . 10  |-  ( ( x  C_  A  /\  Lim  A )  ->  x  C_  On )
62 onssnum 7683 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  x  C_  On )  ->  x  e.  dom  card )
6356, 61, 62sylancr 644 . . . . . . . . 9  |-  ( ( x  C_  A  /\  Lim  A )  ->  x  e.  dom  card )
64 cardid2 7602 . . . . . . . . 9  |-  ( x  e.  dom  card  ->  (
card `  x )  ~~  x )
6563, 64syl 15 . . . . . . . 8  |-  ( ( x  C_  A  /\  Lim  A )  ->  ( card `  x )  ~~  x )
66 ensym 6926 . . . . . . . 8  |-  ( (
card `  x )  ~~  x  ->  x  ~~  ( card `  x )
)
6765, 66syl 15 . . . . . . 7  |-  ( ( x  C_  A  /\  Lim  A )  ->  x  ~~  ( card `  x
) )
6854, 55, 67syl2anc 642 . . . . . 6  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  ~~  ( card `  x ) )
69 simprr 733 . . . . . 6  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  ( cf `  A
)  =  ( card `  x ) )
7068, 69breqtrrd 4065 . . . . 5  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  ~~  ( cf `  A ) )
7151simprd 449 . . . . 5  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  U. x  =  A )
7254, 70, 713jca 1132 . . . 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 ) )
7372ex 423 . . 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 )
) )
7473eximdv 1612 . 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 )
) )
7549, 74mpd 14 1  |-  ( Lim 
A  ->  E. x
( x  C_  A  /\  x  ~~  ( cf `  A )  /\  U. x  =  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   E.wrex 2557   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   U.cuni 3843   |^|cint 3878   |^|_ciin 3922   class class class wbr 4039   Ord word 4407   Oncon0 4408   Lim wlim 4409   dom cdm 4705   ` cfv 5271    ~~ cen 6876   cardccrd 7584   cfccf 7586
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-er 6676  df-en 6880  df-dom 6881  df-card 7588  df-cf 7590
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