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Theorem cfslb 8647
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 5835 . . 3  |-  ( card `  B )  e.  _V
2 ssid 3426 . . . . . . 7  |-  ( card `  B )  C_  ( card `  B )
3 cfslb.1 . . . . . . . . . . 11  |-  A  e. 
_V
43ssex 4511 . . . . . . . . . 10  |-  ( B 
C_  A  ->  B  e.  _V )
54ad2antrr 730 . . . . . . . . 9  |-  ( ( ( B  C_  A  /\  U. B  =  A )  /\  ( card `  B )  C_  ( card `  B ) )  ->  B  e.  _V )
6 selpw 3931 . . . . . . . . . . . . 13  |-  ( x  e.  ~P A  <->  x  C_  A
)
7 sseq1 3428 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  (
x  C_  A  <->  B  C_  A
) )
86, 7syl5bb 260 . . . . . . . . . . . 12  |-  ( x  =  B  ->  (
x  e.  ~P A  <->  B 
C_  A ) )
9 unieq 4170 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  U. x  =  U. B )
109eqeq1d 2430 . . . . . . . . . . . 12  |-  ( x  =  B  ->  ( U. x  =  A  <->  U. B  =  A ) )
118, 10anbi12d 715 . . . . . . . . . . 11  |-  ( x  =  B  ->  (
( x  e.  ~P A  /\  U. x  =  A )  <->  ( B  C_  A  /\  U. B  =  A ) ) )
12 fveq2 5825 . . . . . . . . . . . 12  |-  ( x  =  B  ->  ( card `  x )  =  ( card `  B
) )
1312sseq1d 3434 . . . . . . . . . . 11  |-  ( x  =  B  ->  (
( card `  x )  C_  ( card `  B
)  <->  ( card `  B
)  C_  ( card `  B ) ) )
1411, 13anbi12d 715 . . . . . . . . . 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 3110 . . . . . . . . 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 37 . . . . . . . 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 2720 . . . . . . . . 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 2944 . . . . . . . . . . 11  |-  ( x  e.  { x  e. 
~P A  |  U. x  =  A }  <->  ( x  e.  ~P A  /\  U. x  =  A ) )
1918anbi1i 699 . . . . . . . . . 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 1712 . . . . . . . . 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 252 . . . . . . . 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 215 . . . . . . 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 675 . . . . . 6  |-  ( ( B  C_  A  /\  U. B  =  A )  ->  E. x  e.  {
x  e.  ~P A  |  U. x  =  A }  ( card `  x
)  C_  ( card `  B ) )
24 iinss 4293 . . . . . 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 17 . . . . 5  |-  ( ( B  C_  A  /\  U. B  =  A )  ->  |^|_ x  e.  {
x  e.  ~P A  |  U. x  =  A }  ( card `  x
)  C_  ( card `  B ) )
263cflim3 8643 . . . . . 6  |-  ( Lim 
A  ->  ( cf `  A )  =  |^|_ x  e.  { x  e. 
~P A  |  U. x  =  A } 
( card `  x )
)
2726sseq1d 3434 . . . . 5  |-  ( Lim 
A  ->  ( ( cf `  A )  C_  ( card `  B )  <->  |^|_
x  e.  { x  e.  ~P A  |  U. x  =  A } 
( card `  x )  C_  ( card `  B
) ) )
2825, 27syl5ibr 224 . . . 4  |-  ( Lim 
A  ->  ( ( B  C_  A  /\  U. B  =  A )  ->  ( cf `  A
)  C_  ( card `  B ) ) )
29283impib 1203 . . 3  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  C_  ( card `  B )
)
30 ssdomg 7569 . . 3  |-  ( (
card `  B )  e.  _V  ->  ( ( cf `  A )  C_  ( card `  B )  ->  ( cf `  A
)  ~<_  ( card `  B
) ) )
311, 29, 30mpsyl 65 . 2  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  ~<_  (
card `  B )
)
324adantl 467 . . . . 5  |-  ( ( Lim  A  /\  B  C_  A )  ->  B  e.  _V )
33 limord 5444 . . . . . . 7  |-  ( Lim 
A  ->  Ord  A )
34 ordsson 6574 . . . . . . 7  |-  ( Ord 
A  ->  A  C_  On )
3533, 34syl 17 . . . . . 6  |-  ( Lim 
A  ->  A  C_  On )
36 sstr2 3414 . . . . . 6  |-  ( B 
C_  A  ->  ( A  C_  On  ->  B  C_  On ) )
3735, 36mpan9 471 . . . . 5  |-  ( ( Lim  A  /\  B  C_  A )  ->  B  C_  On )
38 onssnum 8422 . . . . 5  |-  ( ( B  e.  _V  /\  B  C_  On )  ->  B  e.  dom  card )
3932, 37, 38syl2anc 665 . . . 4  |-  ( ( Lim  A  /\  B  C_  A )  ->  B  e.  dom  card )
40 cardid2 8339 . . . 4  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
4139, 40syl 17 . . 3  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( card `  B )  ~~  B )
42413adant3 1025 . 2  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( card `  B )  ~~  B )
43 domentr 7582 . 2  |-  ( ( ( cf `  A
)  ~<_  ( card `  B
)  /\  ( card `  B )  ~~  B
)  ->  ( cf `  A )  ~<_  B )
4431, 42, 43syl2anc 665 1  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  ~<_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872   E.wrex 2715   {crab 2718   _Vcvv 3022    C_ wss 3379   ~Pcpw 3924   U.cuni 4162   |^|_ciin 4243   class class class wbr 4366   dom cdm 4796   Ord word 5384   Oncon0 5385   Lim wlim 5386   ` cfv 5544    ~~ cen 7521    ~<_ cdom 7522   cardccrd 8321   cfccf 8323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-iin 4245  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-se 4756  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-isom 5553  df-riota 6211  df-wrecs 6983  df-recs 7045  df-er 7318  df-en 7525  df-dom 7526  df-card 8325  df-cf 8327
This theorem is referenced by:  cfslbn  8648  cfslb2n  8649  rankcf  9153
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