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Theorem cf0 7872
Description: Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102. (Contributed by NM, 16-Apr-2004.)
Assertion
Ref Expression
cf0  |-  ( cf `  (/) )  =  (/)
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem cf0
StepHypRef Expression
1 cfub 7870 . . 3  |-  ( cf `  (/) )  C_  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }
2 0ss 3484 . . . . . . . . . . . . 13  |-  (/)  C_  U. y
32biantru 493 . . . . . . . . . . . 12  |-  ( y 
C_  (/)  <->  ( y  C_  (/) 
/\  (/)  C_  U. y
) )
4 ss0b 3485 . . . . . . . . . . . 12  |-  ( y 
C_  (/)  <->  y  =  (/) )
53, 4bitr3i 244 . . . . . . . . . . 11  |-  ( ( y  C_  (/)  /\  (/)  C_  U. y
)  <->  y  =  (/) )
65anbi2i 677 . . . . . . . . . 10  |-  ( ( x  =  ( card `  y )  /\  (
y  C_  (/)  /\  (/)  C_  U. y
) )  <->  ( x  =  ( card `  y
)  /\  y  =  (/) ) )
7 ancom 439 . . . . . . . . . 10  |-  ( ( x  =  ( card `  y )  /\  y  =  (/) )  <->  ( y  =  (/)  /\  x  =  ( card `  y
) ) )
86, 7bitri 242 . . . . . . . . 9  |-  ( ( x  =  ( card `  y )  /\  (
y  C_  (/)  /\  (/)  C_  U. y
) )  <->  ( y  =  (/)  /\  x  =  ( card `  y
) ) )
98exbii 1570 . . . . . . . 8  |-  ( E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) )  <->  E. y
( y  =  (/)  /\  x  =  ( card `  y ) ) )
10 0ex 4151 . . . . . . . . . 10  |-  (/)  e.  _V
11 fveq2 5485 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  ( card `  y )  =  (
card `  (/) ) )
1211eqeq2d 2295 . . . . . . . . . 10  |-  ( y  =  (/)  ->  ( x  =  ( card `  y
)  <->  x  =  ( card `  (/) ) ) )
1310, 12ceqsexv 2824 . . . . . . . . 9  |-  ( E. y ( y  =  (/)  /\  x  =  (
card `  y )
)  <->  x  =  ( card `  (/) ) )
14 card0 7586 . . . . . . . . . 10  |-  ( card `  (/) )  =  (/)
1514eqeq2i 2294 . . . . . . . . 9  |-  ( x  =  ( card `  (/) )  <->  x  =  (/) )
1613, 15bitri 242 . . . . . . . 8  |-  ( E. y ( y  =  (/)  /\  x  =  (
card `  y )
)  <->  x  =  (/) )
179, 16bitri 242 . . . . . . 7  |-  ( E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) )  <->  x  =  (/) )
1817abbii 2396 . . . . . 6  |-  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  { x  |  x  =  (/) }
19 df-sn 3647 . . . . . 6  |-  { (/) }  =  { x  |  x  =  (/) }
2018, 19eqtr4i 2307 . . . . 5  |-  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  { (/) }
2120inteqi 3867 . . . 4  |-  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  |^| { (/) }
2210intsn 3899 . . . 4  |-  |^| { (/) }  =  (/)
2321, 22eqtri 2304 . . 3  |-  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  (/)
241, 23sseqtri 3211 . 2  |-  ( cf `  (/) )  C_  (/)
25 ss0b 3485 . 2  |-  ( ( cf `  (/) )  C_  (/)  <->  ( cf `  (/) )  =  (/) )
2624, 25mpbi 201 1  |-  ( cf `  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 360   E.wex 1529    = wceq 1624   {cab 2270    C_ wss 3153   (/)c0 3456   {csn 3641   U.cuni 3828   |^|cint 3863   ` cfv 5221   cardccrd 7563   cfccf 7565
This theorem is referenced by:  cfeq0  7877  cflim2  7884  cfidm  7896  alephsing  7897  alephreg  8199  pwcfsdom  8200  rankcf  8394
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  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 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  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 6859  df-card 7567  df-cf 7569
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