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Theorem cf0 7877
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 `  (/) )  =  (/)

Proof of Theorem cf0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfub 7875 . . 3  |-  ( cf `  (/) )  C_  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }
2 0ss 3483 . . . . . . . . . . . . 13  |-  (/)  C_  U. y
32biantru 491 . . . . . . . . . . . 12  |-  ( y 
C_  (/)  <->  ( y  C_  (/) 
/\  (/)  C_  U. y
) )
4 ss0b 3484 . . . . . . . . . . . 12  |-  ( y 
C_  (/)  <->  y  =  (/) )
53, 4bitr3i 242 . . . . . . . . . . 11  |-  ( ( y  C_  (/)  /\  (/)  C_  U. y
)  <->  y  =  (/) )
65anbi2i 675 . . . . . . . . . 10  |-  ( ( x  =  ( card `  y )  /\  (
y  C_  (/)  /\  (/)  C_  U. y
) )  <->  ( x  =  ( card `  y
)  /\  y  =  (/) ) )
7 ancom 437 . . . . . . . . . 10  |-  ( ( x  =  ( card `  y )  /\  y  =  (/) )  <->  ( y  =  (/)  /\  x  =  ( card `  y
) ) )
86, 7bitri 240 . . . . . . . . 9  |-  ( ( x  =  ( card `  y )  /\  (
y  C_  (/)  /\  (/)  C_  U. y
) )  <->  ( y  =  (/)  /\  x  =  ( card `  y
) ) )
98exbii 1569 . . . . . . . 8  |-  ( E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) )  <->  E. y
( y  =  (/)  /\  x  =  ( card `  y ) ) )
10 0ex 4150 . . . . . . . . . 10  |-  (/)  e.  _V
11 fveq2 5525 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  ( card `  y )  =  (
card `  (/) ) )
1211eqeq2d 2294 . . . . . . . . . 10  |-  ( y  =  (/)  ->  ( x  =  ( card `  y
)  <->  x  =  ( card `  (/) ) ) )
1310, 12ceqsexv 2823 . . . . . . . . 9  |-  ( E. y ( y  =  (/)  /\  x  =  (
card `  y )
)  <->  x  =  ( card `  (/) ) )
14 card0 7591 . . . . . . . . . 10  |-  ( card `  (/) )  =  (/)
1514eqeq2i 2293 . . . . . . . . 9  |-  ( x  =  ( card `  (/) )  <->  x  =  (/) )
1613, 15bitri 240 . . . . . . . 8  |-  ( E. y ( y  =  (/)  /\  x  =  (
card `  y )
)  <->  x  =  (/) )
179, 16bitri 240 . . . . . . 7  |-  ( E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) )  <->  x  =  (/) )
1817abbii 2395 . . . . . 6  |-  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  { x  |  x  =  (/) }
19 df-sn 3646 . . . . . 6  |-  { (/) }  =  { x  |  x  =  (/) }
2018, 19eqtr4i 2306 . . . . 5  |-  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  { (/) }
2120inteqi 3866 . . . 4  |-  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  |^| { (/) }
2210intsn 3898 . . . 4  |-  |^| { (/) }  =  (/)
2321, 22eqtri 2303 . . 3  |-  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  (/)
241, 23sseqtri 3210 . 2  |-  ( cf `  (/) )  C_  (/)
25 ss0b 3484 . 2  |-  ( ( cf `  (/) )  C_  (/)  <->  ( cf `  (/) )  =  (/) )
2624, 25mpbi 199 1  |-  ( cf `  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623   {cab 2269    C_ wss 3152   (/)c0 3455   {csn 3640   U.cuni 3827   |^|cint 3862   ` cfv 5255   cardccrd 7568   cfccf 7570
This theorem is referenced by:  cfeq0  7882  cflim2  7889  cfidm  7901  alephsing  7902  alephreg  8204  pwcfsdom  8205  rankcf  8399
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-en 6864  df-card 7572  df-cf 7574
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