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Theorem cflecard 7879
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 7873 . . 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 3646 . . . . . 6  |-  { (
card `  A ) }  =  { x  |  x  =  ( card `  A ) }
3 ssid 3197 . . . . . . . . 9  |-  A  C_  A
4 ssid 3197 . . . . . . . . . . 11  |-  z  C_  z
5 sseq2 3200 . . . . . . . . . . . 12  |-  ( w  =  z  ->  (
z  C_  w  <->  z  C_  z ) )
65rspcev 2884 . . . . . . . . . . 11  |-  ( ( z  e.  A  /\  z  C_  z )  ->  E. w  e.  A  z  C_  w )
74, 6mpan2 652 . . . . . . . . . 10  |-  ( z  e.  A  ->  E. w  e.  A  z  C_  w )
87rgen 2608 . . . . . . . . 9  |-  A. z  e.  A  E. w  e.  A  z  C_  w
93, 8pm3.2i 441 . . . . . . . 8  |-  ( A 
C_  A  /\  A. z  e.  A  E. w  e.  A  z  C_  w )
10 fveq2 5525 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( card `  y )  =  ( card `  A
) )
1110eqeq2d 2294 . . . . . . . . . 10  |-  ( y  =  A  ->  (
x  =  ( card `  y )  <->  x  =  ( card `  A )
) )
12 sseq1 3199 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
y  C_  A  <->  A  C_  A
) )
13 rexeq 2737 . . . . . . . . . . . 12  |-  ( y  =  A  ->  ( E. w  e.  y 
z  C_  w  <->  E. w  e.  A  z  C_  w ) )
1413ralbidv 2563 . . . . . . . . . . 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 691 . . . . . . . . . 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 691 . . . . . . . . 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 2869 . . . . . . . 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 658 . . . . . . 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 3246 . . . . . 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 3222 . . . . 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 3883 . . . . 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 15 . . . 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 5539 . . . . 5  |-  ( card `  A )  e.  _V
2423intsn 3898 . . . 4  |-  |^| { (
card `  A ) }  =  ( card `  A )
2522, 24syl6sseq 3224 . . 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 3212 . 2  |-  ( A  e.  On  ->  ( cf `  A )  C_  ( card `  A )
)
27 0ss 3483 . . 3  |-  (/)  C_  ( card `  A )
28 cff 7874 . . . . . . 7  |-  cf : On
--> On
2928fdmi 5394 . . . . . 6  |-  dom  cf  =  On
3029eleq2i 2347 . . . . 5  |-  ( A  e.  dom  cf  <->  A  e.  On )
31 ndmfv 5552 . . . . 5  |-  ( -.  A  e.  dom  cf  ->  ( cf `  A
)  =  (/) )
3230, 31sylnbir 298 . . . 4  |-  ( -.  A  e.  On  ->  ( cf `  A )  =  (/) )
3332sseq1d 3205 . . 3  |-  ( -.  A  e.  On  ->  ( ( cf `  A
)  C_  ( card `  A )  <->  (/)  C_  ( card `  A ) ) )
3427, 33mpbiri 224 . 2  |-  ( -.  A  e.  On  ->  ( cf `  A ) 
C_  ( card `  A
) )
3526, 34pm2.61i 156 1  |-  ( cf `  A )  C_  ( card `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   {csn 3640   |^|cint 3862   Oncon0 4392   dom cdm 4689   ` cfv 5255   cardccrd 7568   cfccf 7570
This theorem is referenced by:  cfle  7880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-card 7572  df-cf 7574
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