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Theorem cfle 7876
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 7875 . . 3  |-  ( cf `  A )  C_  ( card `  A )
2 cardonle 7586 . . 3  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
31, 2syl5ss 3191 . 2  |-  ( A  e.  On  ->  ( cf `  A )  C_  A )
4 0ss 3484 . . 3  |-  (/)  C_  A
5 cff 7870 . . . . . . 7  |-  cf : On
--> On
65fdmi 5360 . . . . . 6  |-  dom  cf  =  On
76eleq2i 2348 . . . . 5  |-  ( A  e.  dom  cf  <->  A  e.  On )
8 ndmfv 5514 . . . . 5  |-  ( -.  A  e.  dom  cf  ->  ( cf `  A
)  =  (/) )
97, 8sylnbir 298 . . . 4  |-  ( -.  A  e.  On  ->  ( cf `  A )  =  (/) )
109sseq1d 3206 . . 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 1685    C_ wss 3153   (/)c0 3456   Oncon0 4391    dom cdm 4688   ` cfv 5221   cardccrd 7564   cfccf 7566
This theorem is referenced by:  cfom  7886  cfidm  7897  alephreg  8200  winafp  8315  tskcard  8399  gruina  8436
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-en 6860  df-card 7568  df-cf 7570
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