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Theorem cfle 7882
Description: Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102. (Contributed by NM, 26-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
cfle  |-  ( cf `  A )  C_  A

Proof of Theorem cfle
StepHypRef Expression
1 cflecard 7881 . . 3  |-  ( cf `  A )  C_  ( card `  A )
2 cardonle 7592 . . 3  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
31, 2syl5ss 3192 . 2  |-  ( A  e.  On  ->  ( cf `  A )  C_  A )
4 0ss 3485 . . 3  |-  (/)  C_  A
5 cff 7876 . . . . . . 7  |-  cf : On
--> On
65fdmi 5396 . . . . . 6  |-  dom  cf  =  On
76eleq2i 2349 . . . . 5  |-  ( A  e.  dom  cf  <->  A  e.  On )
8 ndmfv 5554 . . . . 5  |-  ( -.  A  e.  dom  cf  ->  ( cf `  A
)  =  (/) )
97, 8sylnbir 298 . . . 4  |-  ( -.  A  e.  On  ->  ( cf `  A )  =  (/) )
109sseq1d 3207 . . 3  |-  ( -.  A  e.  On  ->  ( ( cf `  A
)  C_  A  <->  (/)  C_  A
) )
114, 10mpbiri 224 . 2  |-  ( -.  A  e.  On  ->  ( cf `  A ) 
C_  A )
123, 11pm2.61i 156 1  |-  ( cf `  A )  C_  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1625    e. wcel 1686    C_ wss 3154   (/)c0 3457   Oncon0 4394   dom cdm 4691   ` cfv 5257   cardccrd 7570   cfccf 7572
This theorem is referenced by:  cfom  7892  cfidm  7903  alephreg  8206  winafp  8321  tskcard  8405  gruina  8442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-en 6866  df-card 7574  df-cf 7576
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