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Theorem cfle 7880
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 7879 . . 3  |-  ( cf `  A )  C_  ( card `  A )
2 cardonle 7590 . . 3  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
31, 2syl5ss 3190 . 2  |-  ( A  e.  On  ->  ( cf `  A )  C_  A )
4 0ss 3483 . . 3  |-  (/)  C_  A
5 cff 7874 . . . . . . 7  |-  cf : On
--> On
65fdmi 5394 . . . . . 6  |-  dom  cf  =  On
76eleq2i 2347 . . . . 5  |-  ( A  e.  dom  cf  <->  A  e.  On )
8 ndmfv 5552 . . . . 5  |-  ( -.  A  e.  dom  cf  ->  ( cf `  A
)  =  (/) )
97, 8sylnbir 298 . . . 4  |-  ( -.  A  e.  On  ->  ( cf `  A )  =  (/) )
109sseq1d 3205 . . 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 1623    e. wcel 1684    C_ wss 3152   (/)c0 3455   Oncon0 4392   dom cdm 4689   ` cfv 5255   cardccrd 7568   cfccf 7570
This theorem is referenced by:  cfom  7890  cfidm  7901  alephreg  8204  winafp  8319  tskcard  8403  gruina  8440
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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-en 6864  df-card 7572  df-cf 7574
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