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Theorem cf0 8120
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 8118 . . 3  |-  ( cf `  (/) )  C_  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }
2 0ss 3648 . . . . . . . . . . . . 13  |-  (/)  C_  U. y
32biantru 492 . . . . . . . . . . . 12  |-  ( y 
C_  (/)  <->  ( y  C_  (/) 
/\  (/)  C_  U. y
) )
4 ss0b 3649 . . . . . . . . . . . 12  |-  ( y 
C_  (/)  <->  y  =  (/) )
53, 4bitr3i 243 . . . . . . . . . . 11  |-  ( ( y  C_  (/)  /\  (/)  C_  U. y
)  <->  y  =  (/) )
65anbi2i 676 . . . . . . . . . 10  |-  ( ( x  =  ( card `  y )  /\  (
y  C_  (/)  /\  (/)  C_  U. y
) )  <->  ( x  =  ( card `  y
)  /\  y  =  (/) ) )
7 ancom 438 . . . . . . . . . 10  |-  ( ( x  =  ( card `  y )  /\  y  =  (/) )  <->  ( y  =  (/)  /\  x  =  ( card `  y
) ) )
86, 7bitri 241 . . . . . . . . 9  |-  ( ( x  =  ( card `  y )  /\  (
y  C_  (/)  /\  (/)  C_  U. y
) )  <->  ( y  =  (/)  /\  x  =  ( card `  y
) ) )
98exbii 1592 . . . . . . . 8  |-  ( E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) )  <->  E. y
( y  =  (/)  /\  x  =  ( card `  y ) ) )
10 0ex 4331 . . . . . . . . . 10  |-  (/)  e.  _V
11 fveq2 5719 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  ( card `  y )  =  (
card `  (/) ) )
1211eqeq2d 2446 . . . . . . . . . 10  |-  ( y  =  (/)  ->  ( x  =  ( card `  y
)  <->  x  =  ( card `  (/) ) ) )
1310, 12ceqsexv 2983 . . . . . . . . 9  |-  ( E. y ( y  =  (/)  /\  x  =  (
card `  y )
)  <->  x  =  ( card `  (/) ) )
14 card0 7834 . . . . . . . . . 10  |-  ( card `  (/) )  =  (/)
1514eqeq2i 2445 . . . . . . . . 9  |-  ( x  =  ( card `  (/) )  <->  x  =  (/) )
1613, 15bitri 241 . . . . . . . 8  |-  ( E. y ( y  =  (/)  /\  x  =  (
card `  y )
)  <->  x  =  (/) )
179, 16bitri 241 . . . . . . 7  |-  ( E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) )  <->  x  =  (/) )
1817abbii 2547 . . . . . 6  |-  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  { x  |  x  =  (/) }
19 df-sn 3812 . . . . . 6  |-  { (/) }  =  { x  |  x  =  (/) }
2018, 19eqtr4i 2458 . . . . 5  |-  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  { (/) }
2120inteqi 4046 . . . 4  |-  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  |^| { (/) }
2210intsn 4078 . . . 4  |-  |^| { (/) }  =  (/)
2321, 22eqtri 2455 . . 3  |-  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  (/)
241, 23sseqtri 3372 . 2  |-  ( cf `  (/) )  C_  (/)
25 ss0b 3649 . 2  |-  ( ( cf `  (/) )  C_  (/)  <->  ( cf `  (/) )  =  (/) )
2624, 25mpbi 200 1  |-  ( cf `  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652   {cab 2421    C_ wss 3312   (/)c0 3620   {csn 3806   U.cuni 4007   |^|cint 4042   ` cfv 5445   cardccrd 7811   cfccf 7813
This theorem is referenced by:  cfeq0  8125  cflim2  8132  cfidm  8144  alephsing  8145  alephreg  8446  pwcfsdom  8447  rankcf  8641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-en 7101  df-card 7815  df-cf 7817
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