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Theorem cflecard 8068
Description: Cofinality is bounded by the cardinality of its argument. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
cflecard  |-  ( cf `  A )  C_  ( card `  A )

Proof of Theorem cflecard
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfval 8062 . . 3  |-  ( A  e.  On  ->  ( cf `  A )  = 
|^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } )
2 df-sn 3765 . . . . . 6  |-  { (
card `  A ) }  =  { x  |  x  =  ( card `  A ) }
3 ssid 3312 . . . . . . . . 9  |-  A  C_  A
4 ssid 3312 . . . . . . . . . . 11  |-  z  C_  z
5 sseq2 3315 . . . . . . . . . . . 12  |-  ( w  =  z  ->  (
z  C_  w  <->  z  C_  z ) )
65rspcev 2997 . . . . . . . . . . 11  |-  ( ( z  e.  A  /\  z  C_  z )  ->  E. w  e.  A  z  C_  w )
74, 6mpan2 653 . . . . . . . . . 10  |-  ( z  e.  A  ->  E. w  e.  A  z  C_  w )
87rgen 2716 . . . . . . . . 9  |-  A. z  e.  A  E. w  e.  A  z  C_  w
93, 8pm3.2i 442 . . . . . . . 8  |-  ( A 
C_  A  /\  A. z  e.  A  E. w  e.  A  z  C_  w )
10 fveq2 5670 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( card `  y )  =  ( card `  A
) )
1110eqeq2d 2400 . . . . . . . . . 10  |-  ( y  =  A  ->  (
x  =  ( card `  y )  <->  x  =  ( card `  A )
) )
12 sseq1 3314 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
y  C_  A  <->  A  C_  A
) )
13 rexeq 2850 . . . . . . . . . . . 12  |-  ( y  =  A  ->  ( E. w  e.  y 
z  C_  w  <->  E. w  e.  A  z  C_  w ) )
1413ralbidv 2671 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( A. z  e.  A  E. w  e.  y 
z  C_  w  <->  A. z  e.  A  E. w  e.  A  z  C_  w ) )
1512, 14anbi12d 692 . . . . . . . . . 10  |-  ( y  =  A  ->  (
( y  C_  A  /\  A. z  e.  A  E. w  e.  y 
z  C_  w )  <->  ( A  C_  A  /\  A. z  e.  A  E. w  e.  A  z  C_  w ) ) )
1611, 15anbi12d 692 . . . . . . . . 9  |-  ( y  =  A  ->  (
( x  =  (
card `  y )  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y 
z  C_  w )
)  <->  ( x  =  ( card `  A
)  /\  ( A  C_  A  /\  A. z  e.  A  E. w  e.  A  z  C_  w ) ) ) )
1716spcegv 2982 . . . . . . . 8  |-  ( A  e.  On  ->  (
( x  =  (
card `  A )  /\  ( A  C_  A  /\  A. z  e.  A  E. w  e.  A  z  C_  w ) )  ->  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) ) )
189, 17mpan2i 659 . . . . . . 7  |-  ( A  e.  On  ->  (
x  =  ( card `  A )  ->  E. y
( x  =  (
card `  y )  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y 
z  C_  w )
) ) )
1918ss2abdv 3361 . . . . . 6  |-  ( A  e.  On  ->  { x  |  x  =  ( card `  A ) } 
C_  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } )
202, 19syl5eqss 3337 . . . . 5  |-  ( A  e.  On  ->  { (
card `  A ) }  C_  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } )
21 intss 4015 . . . . 5  |-  ( { ( card `  A
) }  C_  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) }  ->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } 
C_  |^| { ( card `  A ) } )
2220, 21syl 16 . . . 4  |-  ( A  e.  On  ->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } 
C_  |^| { ( card `  A ) } )
23 fvex 5684 . . . . 5  |-  ( card `  A )  e.  _V
2423intsn 4030 . . . 4  |-  |^| { (
card `  A ) }  =  ( card `  A )
2522, 24syl6sseq 3339 . . 3  |-  ( A  e.  On  ->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } 
C_  ( card `  A
) )
261, 25eqsstrd 3327 . 2  |-  ( A  e.  On  ->  ( cf `  A )  C_  ( card `  A )
)
27 cff 8063 . . . . . 6  |-  cf : On
--> On
2827fdmi 5538 . . . . 5  |-  dom  cf  =  On
2928eleq2i 2453 . . . 4  |-  ( A  e.  dom  cf  <->  A  e.  On )
30 ndmfv 5697 . . . 4  |-  ( -.  A  e.  dom  cf  ->  ( cf `  A
)  =  (/) )
3129, 30sylnbir 299 . . 3  |-  ( -.  A  e.  On  ->  ( cf `  A )  =  (/) )
32 0ss 3601 . . 3  |-  (/)  C_  ( card `  A )
3331, 32syl6eqss 3343 . 2  |-  ( -.  A  e.  On  ->  ( cf `  A ) 
C_  ( card `  A
) )
3426, 33pm2.61i 158 1  |-  ( cf `  A )  C_  ( card `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2375   A.wral 2651   E.wrex 2652    C_ wss 3265   (/)c0 3573   {csn 3759   |^|cint 3994   Oncon0 4524   dom cdm 4820   ` cfv 5396   cardccrd 7757   cfccf 7759
This theorem is referenced by:  cfle  8069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-card 7761  df-cf 7763
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