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Theorem cfslb 7908
Description: Any cofinal subset of  A is at least as large as  ( cf `  A ). (Contributed by Mario Carneiro, 24-Jun-2013.)
Hypothesis
Ref Expression
cfslb.1  |-  A  e. 
_V
Assertion
Ref Expression
cfslb  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  ~<_  B )

Proof of Theorem cfslb
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fvex 5555 . . 3  |-  ( card `  B )  e.  _V
2 ssid 3210 . . . . . . 7  |-  ( card `  B )  C_  ( card `  B )
3 cfslb.1 . . . . . . . . . . 11  |-  A  e. 
_V
43ssex 4174 . . . . . . . . . 10  |-  ( B 
C_  A  ->  B  e.  _V )
54ad2antrr 706 . . . . . . . . 9  |-  ( ( ( B  C_  A  /\  U. B  =  A )  /\  ( card `  B )  C_  ( card `  B ) )  ->  B  e.  _V )
63elpw2 4191 . . . . . . . . . . . . 13  |-  ( x  e.  ~P A  <->  x  C_  A
)
7 sseq1 3212 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  (
x  C_  A  <->  B  C_  A
) )
86, 7syl5bb 248 . . . . . . . . . . . 12  |-  ( x  =  B  ->  (
x  e.  ~P A  <->  B 
C_  A ) )
9 unieq 3852 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  U. x  =  U. B )
109eqeq1d 2304 . . . . . . . . . . . 12  |-  ( x  =  B  ->  ( U. x  =  A  <->  U. B  =  A ) )
118, 10anbi12d 691 . . . . . . . . . . 11  |-  ( x  =  B  ->  (
( x  e.  ~P A  /\  U. x  =  A )  <->  ( B  C_  A  /\  U. B  =  A ) ) )
12 fveq2 5541 . . . . . . . . . . . 12  |-  ( x  =  B  ->  ( card `  x )  =  ( card `  B
) )
1312sseq1d 3218 . . . . . . . . . . 11  |-  ( x  =  B  ->  (
( card `  x )  C_  ( card `  B
)  <->  ( card `  B
)  C_  ( card `  B ) ) )
1411, 13anbi12d 691 . . . . . . . . . 10  |-  ( x  =  B  ->  (
( ( x  e. 
~P A  /\  U. x  =  A )  /\  ( card `  x
)  C_  ( card `  B ) )  <->  ( ( B  C_  A  /\  U. B  =  A )  /\  ( card `  B
)  C_  ( card `  B ) ) ) )
1514spcegv 2882 . . . . . . . . 9  |-  ( B  e.  _V  ->  (
( ( B  C_  A  /\  U. B  =  A )  /\  ( card `  B )  C_  ( card `  B )
)  ->  E. x
( ( x  e. 
~P A  /\  U. x  =  A )  /\  ( card `  x
)  C_  ( card `  B ) ) ) )
165, 15mpcom 32 . . . . . . . 8  |-  ( ( ( B  C_  A  /\  U. B  =  A )  /\  ( card `  B )  C_  ( card `  B ) )  ->  E. x ( ( x  e.  ~P A  /\  U. x  =  A )  /\  ( card `  x )  C_  ( card `  B ) ) )
17 df-rex 2562 . . . . . . . . 9  |-  ( E. x  e.  { x  e.  ~P A  |  U. x  =  A } 
( card `  x )  C_  ( card `  B
)  <->  E. x ( x  e.  { x  e. 
~P A  |  U. x  =  A }  /\  ( card `  x
)  C_  ( card `  B ) ) )
18 rabid 2729 . . . . . . . . . . 11  |-  ( x  e.  { x  e. 
~P A  |  U. x  =  A }  <->  ( x  e.  ~P A  /\  U. x  =  A ) )
1918anbi1i 676 . . . . . . . . . 10  |-  ( ( x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  ( card `  x
)  C_  ( card `  B ) )  <->  ( (
x  e.  ~P A  /\  U. x  =  A )  /\  ( card `  x )  C_  ( card `  B ) ) )
2019exbii 1572 . . . . . . . . 9  |-  ( E. x ( x  e. 
{ x  e.  ~P A  |  U. x  =  A }  /\  ( card `  x )  C_  ( card `  B )
)  <->  E. x ( ( x  e.  ~P A  /\  U. x  =  A )  /\  ( card `  x )  C_  ( card `  B ) ) )
2117, 20bitri 240 . . . . . . . 8  |-  ( E. x  e.  { x  e.  ~P A  |  U. x  =  A } 
( card `  x )  C_  ( card `  B
)  <->  E. x ( ( x  e.  ~P A  /\  U. x  =  A )  /\  ( card `  x )  C_  ( card `  B ) ) )
2216, 21sylibr 203 . . . . . . 7  |-  ( ( ( B  C_  A  /\  U. B  =  A )  /\  ( card `  B )  C_  ( card `  B ) )  ->  E. x  e.  {
x  e.  ~P A  |  U. x  =  A }  ( card `  x
)  C_  ( card `  B ) )
232, 22mpan2 652 . . . . . 6  |-  ( ( B  C_  A  /\  U. B  =  A )  ->  E. x  e.  {
x  e.  ~P A  |  U. x  =  A }  ( card `  x
)  C_  ( card `  B ) )
24 iinss 3969 . . . . . 6  |-  ( E. x  e.  { x  e.  ~P A  |  U. x  =  A } 
( card `  x )  C_  ( card `  B
)  ->  |^|_ x  e. 
{ x  e.  ~P A  |  U. x  =  A }  ( card `  x )  C_  ( card `  B ) )
2523, 24syl 15 . . . . 5  |-  ( ( B  C_  A  /\  U. B  =  A )  ->  |^|_ x  e.  {
x  e.  ~P A  |  U. x  =  A }  ( card `  x
)  C_  ( card `  B ) )
263cflim3 7904 . . . . . 6  |-  ( Lim 
A  ->  ( cf `  A )  =  |^|_ x  e.  { x  e. 
~P A  |  U. x  =  A } 
( card `  x )
)
2726sseq1d 3218 . . . . 5  |-  ( Lim 
A  ->  ( ( cf `  A )  C_  ( card `  B )  <->  |^|_
x  e.  { x  e.  ~P A  |  U. x  =  A } 
( card `  x )  C_  ( card `  B
) ) )
2825, 27syl5ibr 212 . . . 4  |-  ( Lim 
A  ->  ( ( B  C_  A  /\  U. B  =  A )  ->  ( cf `  A
)  C_  ( card `  B ) ) )
29283impib 1149 . . 3  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  C_  ( card `  B )
)
30 ssdomg 6923 . . 3  |-  ( (
card `  B )  e.  _V  ->  ( ( cf `  A )  C_  ( card `  B )  ->  ( cf `  A
)  ~<_  ( card `  B
) ) )
311, 29, 30mpsyl 59 . 2  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  ~<_  (
card `  B )
)
324adantl 452 . . . . 5  |-  ( ( Lim  A  /\  B  C_  A )  ->  B  e.  _V )
33 limord 4467 . . . . . . 7  |-  ( Lim 
A  ->  Ord  A )
34 ordsson 4597 . . . . . . 7  |-  ( Ord 
A  ->  A  C_  On )
3533, 34syl 15 . . . . . 6  |-  ( Lim 
A  ->  A  C_  On )
36 sstr2 3199 . . . . . 6  |-  ( B 
C_  A  ->  ( A  C_  On  ->  B  C_  On ) )
3735, 36mpan9 455 . . . . 5  |-  ( ( Lim  A  /\  B  C_  A )  ->  B  C_  On )
38 onssnum 7683 . . . . 5  |-  ( ( B  e.  _V  /\  B  C_  On )  ->  B  e.  dom  card )
3932, 37, 38syl2anc 642 . . . 4  |-  ( ( Lim  A  /\  B  C_  A )  ->  B  e.  dom  card )
40 cardid2 7602 . . . 4  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
4139, 40syl 15 . . 3  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( card `  B )  ~~  B )
42413adant3 975 . 2  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( card `  B )  ~~  B )
43 domentr 6936 . 2  |-  ( ( ( cf `  A
)  ~<_  ( card `  B
)  /\  ( card `  B )  ~~  B
)  ->  ( cf `  A )  ~<_  B )
4431, 42, 43syl2anc 642 1  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   E.wrex 2557   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   |^|_ciin 3922   class class class wbr 4039   Ord word 4407   Oncon0 4408   Lim wlim 4409   dom cdm 4705   ` cfv 5271    ~~ cen 6876    ~<_ cdom 6877   cardccrd 7584   cfccf 7586
This theorem is referenced by:  cfslbn  7909  cfslb2n  7910  rankcf  8415
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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