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Theorem cfle 7848
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 7847 . . 3  |-  ( cf `  A )  C_  ( card `  A )
2 cardonle 7558 . . 3  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
31, 2syl5ss 3165 . 2  |-  ( A  e.  On  ->  ( cf `  A )  C_  A )
4 0ss 3458 . . 3  |-  (/)  C_  A
5 cff 7842 . . . . . . 7  |-  cf : On
--> On
65fdmi 5332 . . . . . 6  |-  dom  cf  =  On
76eleq2i 2322 . . . . 5  |-  ( A  e.  dom  cf  <->  A  e.  On )
8 ndmfv 5486 . . . . 5  |-  ( -.  A  e.  dom  cf  ->  ( cf `  A
)  =  (/) )
97, 8sylnbir 300 . . . 4  |-  ( -.  A  e.  On  ->  ( cf `  A )  =  (/) )
109sseq1d 3180 . . 3  |-  ( -.  A  e.  On  ->  ( ( cf `  A
)  C_  A  <->  (/)  C_  A
) )
114, 10mpbiri 226 . 2  |-  ( -.  A  e.  On  ->  ( cf `  A ) 
C_  A )
123, 11pm2.61i 158 1  |-  ( cf `  A )  C_  A
Colors of variables: wff set class
Syntax hints:   -. wn 5    = wceq 1619    e. wcel 1621    C_ wss 3127   (/)c0 3430   Oncon0 4364   dom cdm 4661   ` cfv 4673   cardccrd 7536   cfccf 7538
This theorem is referenced by:  cfom  7858  cfidm  7869  alephreg  8172  winafp  8287  tskcard  8371  gruina  8408
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-en 6832  df-card 7540  df-cf 7542
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