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Theorem cfslb 8146
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 5742 . . 3  |-  ( card `  B )  e.  _V
2 ssid 3367 . . . . . . 7  |-  ( card `  B )  C_  ( card `  B )
3 cfslb.1 . . . . . . . . . . 11  |-  A  e. 
_V
43ssex 4347 . . . . . . . . . 10  |-  ( B 
C_  A  ->  B  e.  _V )
54ad2antrr 707 . . . . . . . . 9  |-  ( ( ( B  C_  A  /\  U. B  =  A )  /\  ( card `  B )  C_  ( card `  B ) )  ->  B  e.  _V )
63elpw2 4364 . . . . . . . . . . . . 13  |-  ( x  e.  ~P A  <->  x  C_  A
)
7 sseq1 3369 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  (
x  C_  A  <->  B  C_  A
) )
86, 7syl5bb 249 . . . . . . . . . . . 12  |-  ( x  =  B  ->  (
x  e.  ~P A  <->  B 
C_  A ) )
9 unieq 4024 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  U. x  =  U. B )
109eqeq1d 2444 . . . . . . . . . . . 12  |-  ( x  =  B  ->  ( U. x  =  A  <->  U. B  =  A ) )
118, 10anbi12d 692 . . . . . . . . . . 11  |-  ( x  =  B  ->  (
( x  e.  ~P A  /\  U. x  =  A )  <->  ( B  C_  A  /\  U. B  =  A ) ) )
12 fveq2 5728 . . . . . . . . . . . 12  |-  ( x  =  B  ->  ( card `  x )  =  ( card `  B
) )
1312sseq1d 3375 . . . . . . . . . . 11  |-  ( x  =  B  ->  (
( card `  x )  C_  ( card `  B
)  <->  ( card `  B
)  C_  ( card `  B ) ) )
1411, 13anbi12d 692 . . . . . . . . . 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 3037 . . . . . . . . 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 34 . . . . . . . 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 2711 . . . . . . . . 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 2884 . . . . . . . . . . 11  |-  ( x  e.  { x  e. 
~P A  |  U. x  =  A }  <->  ( x  e.  ~P A  /\  U. x  =  A ) )
1918anbi1i 677 . . . . . . . . . 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 1592 . . . . . . . . 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 241 . . . . . . . 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 204 . . . . . . 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 653 . . . . . 6  |-  ( ( B  C_  A  /\  U. B  =  A )  ->  E. x  e.  {
x  e.  ~P A  |  U. x  =  A }  ( card `  x
)  C_  ( card `  B ) )
24 iinss 4142 . . . . . 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 8142 . . . . . 6  |-  ( Lim 
A  ->  ( cf `  A )  =  |^|_ x  e.  { x  e. 
~P A  |  U. x  =  A } 
( card `  x )
)
2726sseq1d 3375 . . . . 5  |-  ( Lim 
A  ->  ( ( cf `  A )  C_  ( card `  B )  <->  |^|_
x  e.  { x  e.  ~P A  |  U. x  =  A } 
( card `  x )  C_  ( card `  B
) ) )
2825, 27syl5ibr 213 . . . 4  |-  ( Lim 
A  ->  ( ( B  C_  A  /\  U. B  =  A )  ->  ( cf `  A
)  C_  ( card `  B ) ) )
29283impib 1151 . . 3  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  C_  ( card `  B )
)
30 ssdomg 7153 . . 3  |-  ( (
card `  B )  e.  _V  ->  ( ( cf `  A )  C_  ( card `  B )  ->  ( cf `  A
)  ~<_  ( card `  B
) ) )
311, 29, 30mpsyl 61 . 2  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  ~<_  (
card `  B )
)
324adantl 453 . . . . 5  |-  ( ( Lim  A  /\  B  C_  A )  ->  B  e.  _V )
33 limord 4640 . . . . . . 7  |-  ( Lim 
A  ->  Ord  A )
34 ordsson 4770 . . . . . . 7  |-  ( Ord 
A  ->  A  C_  On )
3533, 34syl 16 . . . . . 6  |-  ( Lim 
A  ->  A  C_  On )
36 sstr2 3355 . . . . . 6  |-  ( B 
C_  A  ->  ( A  C_  On  ->  B  C_  On ) )
3735, 36mpan9 456 . . . . 5  |-  ( ( Lim  A  /\  B  C_  A )  ->  B  C_  On )
38 onssnum 7921 . . . . 5  |-  ( ( B  e.  _V  /\  B  C_  On )  ->  B  e.  dom  card )
3932, 37, 38syl2anc 643 . . . 4  |-  ( ( Lim  A  /\  B  C_  A )  ->  B  e.  dom  card )
40 cardid2 7840 . . . 4  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
4139, 40syl 16 . . 3  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( card `  B )  ~~  B )
42413adant3 977 . 2  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( card `  B )  ~~  B )
43 domentr 7166 . 2  |-  ( ( ( cf `  A
)  ~<_  ( card `  B
)  /\  ( card `  B )  ~~  B
)  ->  ( cf `  A )  ~<_  B )
4431, 42, 43syl2anc 643 1  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   E.wrex 2706   {crab 2709   _Vcvv 2956    C_ wss 3320   ~Pcpw 3799   U.cuni 4015   |^|_ciin 4094   class class class wbr 4212   Ord word 4580   Oncon0 4581   Lim wlim 4582   dom cdm 4878   ` cfv 5454    ~~ cen 7106    ~<_ cdom 7107   cardccrd 7822   cfccf 7824
This theorem is referenced by:  cfslbn  8147  cfslb2n  8148  rankcf  8652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-riota 6549  df-recs 6633  df-er 6905  df-en 7110  df-dom 7111  df-card 7826  df-cf 7828
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