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Theorem cfle 8090
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 8089 . . 3  |-  ( cf `  A )  C_  ( card `  A )
2 cardonle 7800 . . 3  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
31, 2syl5ss 3319 . 2  |-  ( A  e.  On  ->  ( cf `  A )  C_  A )
4 cff 8084 . . . . . 6  |-  cf : On
--> On
54fdmi 5555 . . . . 5  |-  dom  cf  =  On
65eleq2i 2468 . . . 4  |-  ( A  e.  dom  cf  <->  A  e.  On )
7 ndmfv 5714 . . . 4  |-  ( -.  A  e.  dom  cf  ->  ( cf `  A
)  =  (/) )
86, 7sylnbir 299 . . 3  |-  ( -.  A  e.  On  ->  ( cf `  A )  =  (/) )
9 0ss 3616 . . 3  |-  (/)  C_  A
108, 9syl6eqss 3358 . 2  |-  ( -.  A  e.  On  ->  ( cf `  A ) 
C_  A )
113, 10pm2.61i 158 1  |-  ( cf `  A )  C_  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1721    C_ wss 3280   (/)c0 3588   Oncon0 4541   dom cdm 4837   ` cfv 5413   cardccrd 7778   cfccf 7780
This theorem is referenced by:  cfom  8100  cfidm  8111  alephreg  8413  winafp  8528  tskcard  8612  gruina  8649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-en 7069  df-card 7782  df-cf 7784
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