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Theorem cfle 7896
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 7895 . . 3  |-  ( cf `  A )  C_  ( card `  A )
2 cardonle 7606 . . 3  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
31, 2syl5ss 3203 . 2  |-  ( A  e.  On  ->  ( cf `  A )  C_  A )
4 0ss 3496 . . 3  |-  (/)  C_  A
5 cff 7890 . . . . . . 7  |-  cf : On
--> On
65fdmi 5410 . . . . . 6  |-  dom  cf  =  On
76eleq2i 2360 . . . . 5  |-  ( A  e.  dom  cf  <->  A  e.  On )
8 ndmfv 5568 . . . . 5  |-  ( -.  A  e.  dom  cf  ->  ( cf `  A
)  =  (/) )
97, 8sylnbir 298 . . . 4  |-  ( -.  A  e.  On  ->  ( cf `  A )  =  (/) )
109sseq1d 3218 . . 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 1632    e. wcel 1696    C_ wss 3165   (/)c0 3468   Oncon0 4408   dom cdm 4705   ` cfv 5271   cardccrd 7584   cfccf 7586
This theorem is referenced by:  cfom  7906  cfidm  7917  alephreg  8220  winafp  8335  tskcard  8419  gruina  8456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-en 6880  df-card 7588  df-cf 7590
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