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Theorem cfslb 8550
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 5812 . . 3  |-  ( card `  B )  e.  _V
2 ssid 3486 . . . . . . 7  |-  ( card `  B )  C_  ( card `  B )
3 cfslb.1 . . . . . . . . . . 11  |-  A  e. 
_V
43ssex 4547 . . . . . . . . . 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 3978 . . . . . . . . . . . . 13  |-  ( x  e.  ~P A  <->  x  C_  A
)
7 sseq1 3488 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  (
x  C_  A  <->  B  C_  A
) )
86, 7syl5bb 257 . . . . . . . . . . . 12  |-  ( x  =  B  ->  (
x  e.  ~P A  <->  B 
C_  A ) )
9 unieq 4210 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  U. x  =  U. B )
109eqeq1d 2456 . . . . . . . . . . . 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 5802 . . . . . . . . . . . 12  |-  ( x  =  B  ->  ( card `  x )  =  ( card `  B
) )
1312sseq1d 3494 . . . . . . . . . . 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 3164 . . . . . . . . 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 2805 . . . . . . . . 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 3003 . . . . . . . . . . 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 1635 . . . . . . . . 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 4332 . . . . . 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 8546 . . . . . 6  |-  ( Lim 
A  ->  ( cf `  A )  =  |^|_ x  e.  { x  e. 
~P A  |  U. x  =  A } 
( card `  x )
)
2726sseq1d 3494 . . . . 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 1186 . . 3  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( cf `  A )  C_  ( card `  B )
)
30 ssdomg 7468 . . 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 4889 . . . . . . 7  |-  ( Lim 
A  ->  Ord  A )
34 ordsson 6514 . . . . . . 7  |-  ( Ord 
A  ->  A  C_  On )
3533, 34syl 16 . . . . . 6  |-  ( Lim 
A  ->  A  C_  On )
36 sstr2 3474 . . . . . 6  |-  ( B 
C_  A  ->  ( A  C_  On  ->  B  C_  On ) )
3735, 36mpan9 469 . . . . 5  |-  ( ( Lim  A  /\  B  C_  A )  ->  B  C_  On )
38 onssnum 8325 . . . . 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 8238 . . . 4  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
4139, 40syl 16 . . 3  |-  ( ( Lim  A  /\  B  C_  A )  ->  ( card `  B )  ~~  B )
42413adant3 1008 . 2  |-  ( ( Lim  A  /\  B  C_  A  /\  U. B  =  A )  ->  ( card `  B )  ~~  B )
43 domentr 7481 . 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 965    = wceq 1370   E.wex 1587    e. wcel 1758   E.wrex 2800   {crab 2803   _Vcvv 3078    C_ wss 3439   ~Pcpw 3971   U.cuni 4202   |^|_ciin 4283   class class class wbr 4403   Ord word 4829   Oncon0 4830   Lim wlim 4831   dom cdm 4951   ` cfv 5529    ~~ cen 7420    ~<_ cdom 7421   cardccrd 8220   cfccf 8222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-recs 6945  df-er 7214  df-en 7424  df-dom 7425  df-card 8224  df-cf 8226
This theorem is referenced by:  cfslbn  8551  cfslb2n  8552  rankcf  9059
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