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Theorem cfslb 8663
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 5882 . . 3  |-  ( card `  B )  e.  _V
2 ssid 3518 . . . . . . 7  |-  ( card `  B )  C_  ( card `  B )
3 cfslb.1 . . . . . . . . . . 11  |-  A  e. 
_V
43ssex 4600 . . . . . . . . . 10  |-  ( B 
C_  A  ->  B  e.  _V )
54ad2antrr 725 . . . . . . . . 9  |-  ( ( ( B  C_  A  /\  U. B  =  A )  /\  ( card `  B )  C_  ( card `  B ) )  ->  B  e.  _V )
6 selpw 4022 . . . . . . . . . . . . 13  |-  ( x  e.  ~P A  <->  x  C_  A
)
7 sseq1 3520 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  (
x  C_  A  <->  B  C_  A
) )
86, 7syl5bb 257 . . . . . . . . . . . 12  |-  ( x  =  B  ->  (
x  e.  ~P A  <->  B 
C_  A ) )
9 unieq 4259 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  U. x  =  U. B )
109eqeq1d 2459 . . . . . . . . . . . 12  |-  ( x  =  B  ->  ( U. x  =  A  <->  U. B  =  A ) )
118, 10anbi12d 710 . . . . . . . . . . 11  |-  ( x  =  B  ->  (
( x  e.  ~P A  /\  U. x  =  A )  <->  ( B  C_  A  /\  U. B  =  A ) ) )
12 fveq2 5872 . . . . . . . . . . . 12  |-  ( x  =  B  ->  ( card `  x )  =  ( card `  B
) )
1312sseq1d 3526 . . . . . . . . . . 11  |-  ( x  =  B  ->  (
( card `  x )  C_  ( card `  B
)  <->  ( card `  B
)  C_  ( card `  B ) ) )
1411, 13anbi12d 710 . . . . . . . . . 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 3195 . . . . . . . . 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 36 . . . . . . . 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 2813 . . . . . . . . 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 3034 . . . . . . . . . . 11  |-  ( x  e.  { x  e. 
~P A  |  U. x  =  A }  <->  ( x  e.  ~P A  /\  U. x  =  A ) )
1918anbi1i 695 . . . . . . . . . 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 1668 . . . . . . . . 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 249 . . . . . . . 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 212 . . . . . . 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 671 . . . . . 6  |-  ( ( B  C_  A  /\  U. B  =  A )  ->  E. x  e.  {
x  e.  ~P A  |  U. x  =  A }  ( card `  x
)  C_  ( card `  B ) )
24 iinss 4383 . . . . . 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 16 . . . . 5  |-  ( ( B  C_  A  /\  U. B  =  A )  ->  |^|_ x  e.  {
x  e.  ~P A  |  U. x  =  A }  ( card `  x
)  C_  ( card `  B ) )
263cflim3 8659 . . . . . 6  |-  ( Lim 
A  ->  ( cf `  A )  =  |^|_ x  e.  { x  e. 
~P A  |  U. x  =  A } 
( card `  x )
)
2726sseq1d 3526 . . . . 5  |-  ( Lim 
A  ->  ( ( cf `  A )  C_  ( card `  B )  <->  |^|_
x  e.  { x  e.  ~P A  |  U. x  =  A } 
( card `  x )  C_  ( card `  B
) ) )
2825, 27syl5ibr 221 . . . 4  |-  ( Lim 
A  ->  ( ( B  C_  A  /\  U. B  =  A )  ->  ( cf `  A
)  C_  ( card `  B ) ) )
29283impib 1194 . . 3  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  C_  ( card `  B )
)
30 ssdomg 7580 . . 3  |-  ( (
card `  B )  e.  _V  ->  ( ( cf `  A )  C_  ( card `  B )  ->  ( cf `  A
)  ~<_  ( card `  B
) ) )
311, 29, 30mpsyl 63 . 2  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  ~<_  (
card `  B )
)
324adantl 466 . . . . 5  |-  ( ( Lim  A  /\  B  C_  A )  ->  B  e.  _V )
33 limord 4946 . . . . . . 7  |-  ( Lim 
A  ->  Ord  A )
34 ordsson 6624 . . . . . . 7  |-  ( Ord 
A  ->  A  C_  On )
3533, 34syl 16 . . . . . 6  |-  ( Lim 
A  ->  A  C_  On )
36 sstr2 3506 . . . . . 6  |-  ( B 
C_  A  ->  ( A  C_  On  ->  B  C_  On ) )
3735, 36mpan9 469 . . . . 5  |-  ( ( Lim  A  /\  B  C_  A )  ->  B  C_  On )
38 onssnum 8438 . . . . 5  |-  ( ( B  e.  _V  /\  B  C_  On )  ->  B  e.  dom  card )
3932, 37, 38syl2anc 661 . . . 4  |-  ( ( Lim  A  /\  B  C_  A )  ->  B  e.  dom  card )
40 cardid2 8351 . . . 4  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
4139, 40syl 16 . . 3  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( card `  B )  ~~  B )
42413adant3 1016 . 2  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( card `  B )  ~~  B )
43 domentr 7593 . 2  |-  ( ( ( cf `  A
)  ~<_  ( card `  B
)  /\  ( card `  B )  ~~  B
)  ->  ( cf `  A )  ~<_  B )
4431, 42, 43syl2anc 661 1  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  ~<_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819   E.wrex 2808   {crab 2811   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015   U.cuni 4251   |^|_ciin 4333   class class class wbr 4456   Ord word 4886   Oncon0 4887   Lim wlim 4888   dom cdm 5008   ` cfv 5594    ~~ cen 7532    ~<_ cdom 7533   cardccrd 8333   cfccf 8335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-recs 7060  df-er 7329  df-en 7536  df-dom 7537  df-card 8337  df-cf 8339
This theorem is referenced by:  cfslbn  8664  cfslb2n  8665  rankcf  9172
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