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Theorem cf0 7810
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
StepHypRef Expression
1 cfub 7808 . . 3  |-  ( cf `  (/) )  C_  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }
2 0ss 3425 . . . . . . . . . . . . 13  |-  (/)  C_  U. y
32biantru 493 . . . . . . . . . . . 12  |-  ( y 
C_  (/)  <->  ( y  C_  (/) 
/\  (/)  C_  U. y
) )
4 ss0b 3426 . . . . . . . . . . . 12  |-  ( y 
C_  (/)  <->  y  =  (/) )
53, 4bitr3i 244 . . . . . . . . . . 11  |-  ( ( y  C_  (/)  /\  (/)  C_  U. y
)  <->  y  =  (/) )
65anbi2i 678 . . . . . . . . . 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 1580 . . . . . . . 8  |-  ( E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) )  <->  E. y
( y  =  (/)  /\  x  =  ( card `  y ) ) )
10 0ex 4090 . . . . . . . . . 10  |-  (/)  e.  _V
11 fveq2 5423 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  ( card `  y )  =  (
card `  (/) ) )
1211eqeq2d 2267 . . . . . . . . . 10  |-  ( y  =  (/)  ->  ( x  =  ( card `  y
)  <->  x  =  ( card `  (/) ) ) )
1310, 12ceqsexv 2774 . . . . . . . . 9  |-  ( E. y ( y  =  (/)  /\  x  =  (
card `  y )
)  <->  x  =  ( card `  (/) ) )
14 card0 7524 . . . . . . . . . 10  |-  ( card `  (/) )  =  (/)
1514eqeq2i 2266 . . . . . . . . 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 2368 . . . . . 6  |-  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  { x  |  x  =  (/) }
19 df-sn 3587 . . . . . 6  |-  { (/) }  =  { x  |  x  =  (/) }
2018, 19eqtr4i 2279 . . . . 5  |-  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  { (/) }
2120inteqi 3807 . . . 4  |-  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  |^| { (/) }
2210intsn 3839 . . . 4  |-  |^| { (/) }  =  (/)
2321, 22eqtri 2276 . . 3  |-  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  (/)
241, 23sseqtri 3152 . 2  |-  ( cf `  (/) )  C_  (/)
25 ss0b 3426 . 2  |-  ( ( cf `  (/) )  C_  (/)  <->  ( cf `  (/) )  =  (/) )
2624, 25mpbi 201 1  |-  ( cf `  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 360   E.wex 1537    = wceq 1619   {cab 2242    C_ wss 3094   (/)c0 3397   {csn 3581   U.cuni 3768   |^|cint 3803   ` cfv 4638   cardccrd 7501   cfccf 7503
This theorem is referenced by:  cfeq0  7815  cflim2  7822  cfidm  7834  alephsing  7835  alephreg  8137  pwcfsdom  8138  rankcf  8332
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 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-en 6797  df-card 7505  df-cf 7507
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