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Theorem cfss 8692
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 8689 . . . . 5  |-  ( Lim 
A  ->  ( cf `  A )  =  |^|_ x  e.  { x  e. 
~P A  |  U. x  =  A } 
( card `  x )
)
3 fvex 5873 . . . . . . 7  |-  ( card `  x )  e.  _V
43dfiin2 4312 . . . . . 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 8375 . . . . . . . . . 10  |-  ( card `  x )  e.  On
6 eleq1 2516 . . . . . . . . . 10  |-  ( y  =  ( card `  x
)  ->  ( y  e.  On  <->  ( card `  x
)  e.  On ) )
75, 6mpbiri 237 . . . . . . . . 9  |-  ( y  =  ( card `  x
)  ->  y  e.  On )
87rexlimivw 2875 . . . . . . . 8  |-  ( E. x  e.  { x  e.  ~P A  |  U. x  =  A }
y  =  ( card `  x )  ->  y  e.  On )
98abssi 3503 . . . . . . 7  |-  { y  |  E. x  e. 
{ x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) }  C_  On
10 limuni 5482 . . . . . . . . . . . 12  |-  ( Lim 
A  ->  A  =  U. A )
1110eqcomd 2456 . . . . . . . . . . 11  |-  ( Lim 
A  ->  U. A  =  A )
12 fveq2 5863 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  ( card `  x )  =  ( card `  A
) )
1312eqcomd 2456 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  ( card `  A )  =  ( card `  x
) )
1413biantrud 510 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  ( U. A  =  A  <->  ( U. A  =  A  /\  ( card `  A
)  =  ( card `  x ) ) ) )
15 unieq 4205 . . . . . . . . . . . . . . . 16  |-  ( x  =  A  ->  U. x  =  U. A )
1615eqeq1d 2452 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  ( U. x  =  A  <->  U. A  =  A ) )
171pwid 3964 . . . . . . . . . . . . . . . . 17  |-  A  e. 
~P A
18 eleq1 2516 . . . . . . . . . . . . . . . . 17  |-  ( x  =  A  ->  (
x  e.  ~P A  <->  A  e.  ~P A ) )
1917, 18mpbiri 237 . . . . . . . . . . . . . . . 16  |-  ( x  =  A  ->  x  e.  ~P A )
2019biantrurd 511 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  ( U. x  =  A  <->  ( x  e.  ~P A  /\  U. x  =  A ) ) )
2116, 20bitr3d 259 . . . . . . . . . . . . . 14  |-  ( x  =  A  ->  ( U. A  =  A  <->  ( x  e.  ~P A  /\  U. x  =  A ) ) )
2221anbi1d 710 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  (
( U. A  =  A  /\  ( card `  A )  =  (
card `  x )
)  <->  ( ( x  e.  ~P A  /\  U. x  =  A )  /\  ( card `  A
)  =  ( card `  x ) ) ) )
2314, 22bitr2d 258 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
( ( x  e. 
~P A  /\  U. x  =  A )  /\  ( card `  A
)  =  ( card `  x ) )  <->  U. A  =  A ) )
241, 23spcev 3140 . . . . . . . . . . 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 2742 . . . . . . . . . . 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 2966 . . . . . . . . . . . . 13  |-  ( x  e.  { x  e. 
~P A  |  U. x  =  A }  <->  ( x  e.  ~P A  /\  U. x  =  A ) )
2827anbi1i 700 . . . . . . . . . . . 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 1717 . . . . . . . . . . 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 253 . . . . . . . . . 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 216 . . . . . . . . 9  |-  ( Lim 
A  ->  E. x  e.  { x  e.  ~P A  |  U. x  =  A }  ( card `  A )  =  (
card `  x )
)
32 fvex 5873 . . . . . . . . . 10  |-  ( card `  A )  e.  _V
33 eqeq1 2454 . . . . . . . . . . 11  |-  ( y  =  ( card `  A
)  ->  ( y  =  ( card `  x
)  <->  ( card `  A
)  =  ( card `  x ) ) )
3433rexbidv 2900 . . . . . . . . . 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 3140 . . . . . . . . 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 3750 . . . . . . . 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 216 . . . . . . 7  |-  ( Lim 
A  ->  { y  |  E. x  e.  {
x  e.  ~P A  |  U. x  =  A } y  =  (
card `  x ) }  =/=  (/) )
39 onint 6619 . . . . . . 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 668 . . . . . 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 2532 . . . . 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 2528 . . . 4  |-  ( Lim 
A  ->  ( cf `  A )  e.  {
y  |  E. x  e.  { x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) } )
43 fvex 5873 . . . . 5  |-  ( cf `  A )  e.  _V
44 eqeq1 2454 . . . . . 6  |-  ( y  =  ( cf `  A
)  ->  ( y  =  ( card `  x
)  <->  ( cf `  A
)  =  ( card `  x ) ) )
4544rexbidv 2900 . . . . 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 3184 . . . 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 200 . . 3  |-  ( Lim 
A  ->  E. x  e.  { x  e.  ~P A  |  U. x  =  A }  ( cf `  A )  =  (
card `  x )
)
48 df-rex 2742 . . 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 200 . 2  |-  ( Lim 
A  ->  E. x
( x  e.  {
x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A )  =  (
card `  x )
) )
50 simprl 763 . . . . . . . 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 200 . . . . . . 7  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  ( x  e. 
~P A  /\  U. x  =  A )
)
5251simpld 461 . . . . . 6  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  e.  ~P A )
5352elpwid 3960 . . . . 5  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  C_  A
)
54 simpl 459 . . . . . . 7  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  Lim  A )
55 vex 3047 . . . . . . . . . 10  |-  x  e. 
_V
56 limord 5481 . . . . . . . . . . . 12  |-  ( Lim 
A  ->  Ord  A )
57 ordsson 6613 . . . . . . . . . . . 12  |-  ( Ord 
A  ->  A  C_  On )
5856, 57syl 17 . . . . . . . . . . 11  |-  ( Lim 
A  ->  A  C_  On )
59 sstr 3439 . . . . . . . . . . 11  |-  ( ( x  C_  A  /\  A  C_  On )  ->  x  C_  On )
6058, 59sylan2 477 . . . . . . . . . 10  |-  ( ( x  C_  A  /\  Lim  A )  ->  x  C_  On )
61 onssnum 8468 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  x  C_  On )  ->  x  e.  dom  card )
6255, 60, 61sylancr 668 . . . . . . . . 9  |-  ( ( x  C_  A  /\  Lim  A )  ->  x  e.  dom  card )
63 cardid2 8384 . . . . . . . . 9  |-  ( x  e.  dom  card  ->  (
card `  x )  ~~  x )
6462, 63syl 17 . . . . . . . 8  |-  ( ( x  C_  A  /\  Lim  A )  ->  ( card `  x )  ~~  x )
6564ensymd 7617 . . . . . . 7  |-  ( ( x  C_  A  /\  Lim  A )  ->  x  ~~  ( card `  x
) )
6653, 54, 65syl2anc 666 . . . . . 6  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  ~~  ( card `  x ) )
67 simprr 765 . . . . . 6  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  ( cf `  A
)  =  ( card `  x ) )
6866, 67breqtrrd 4428 . . . . 5  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  x  ~~  ( cf `  A ) )
6951simprd 465 . . . . 5  |-  ( ( Lim  A  /\  (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( cf `  A
)  =  ( card `  x ) ) )  ->  U. x  =  A )
7053, 68, 693jca 1187 . . . 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 436 . . 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 1763 . 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 371    /\ w3a 984    = wceq 1443   E.wex 1662    e. wcel 1886   {cab 2436    =/= wne 2621   E.wrex 2737   {crab 2740   _Vcvv 3044    C_ wss 3403   (/)c0 3730   ~Pcpw 3950   U.cuni 4197   |^|cint 4233   |^|_ciin 4278   class class class wbr 4401   dom cdm 4833   Ord word 5421   Oncon0 5422   Lim wlim 5423   ` cfv 5581    ~~ cen 7563   cardccrd 8366   cfccf 8368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-wrecs 7025  df-recs 7087  df-er 7360  df-en 7567  df-dom 7568  df-card 8370  df-cf 8372
This theorem is referenced by: (None)
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