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Theorem cf0 7761
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 7759 . . 3  |-  ( cf `  (/) )  C_  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }
2 0ss 3390 . . . . . . . . . . . . 13  |-  (/)  C_  U. y
32biantru 493 . . . . . . . . . . . 12  |-  ( y 
C_  (/)  <->  ( y  C_  (/) 
/\  (/)  C_  U. y
) )
4 ss0b 3391 . . . . . . . . . . . 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 4047 . . . . . . . . . 10  |-  (/)  e.  _V
11 fveq2 5377 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  ( card `  y )  =  (
card `  (/) ) )
1211eqeq2d 2264 . . . . . . . . . 10  |-  ( y  =  (/)  ->  ( x  =  ( card `  y
)  <->  x  =  ( card `  (/) ) ) )
1310, 12ceqsexv 2761 . . . . . . . . 9  |-  ( E. y ( y  =  (/)  /\  x  =  (
card `  y )
)  <->  x  =  ( card `  (/) ) )
14 card0 7475 . . . . . . . . . 10  |-  ( card `  (/) )  =  (/)
1514eqeq2i 2263 . . . . . . . . 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 2361 . . . . . 6  |-  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  { x  |  x  =  (/) }
19 df-sn 3550 . . . . . 6  |-  { (/) }  =  { x  |  x  =  (/) }
2018, 19eqtr4i 2276 . . . . 5  |-  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  { (/) }
2120inteqi 3764 . . . 4  |-  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  |^| { (/) }
2210intsn 3796 . . . 4  |-  |^| { (/) }  =  (/)
2321, 22eqtri 2273 . . 3  |-  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  (/)  /\  (/)  C_  U. y
) ) }  =  (/)
241, 23sseqtri 3131 . 2  |-  ( cf `  (/) )  C_  (/)
25 ss0b 3391 . 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 2239    C_ wss 3078   (/)c0 3362   {csn 3544   U.cuni 3727   |^|cint 3760   ` cfv 4592   cardccrd 7452   cfccf 7454
This theorem is referenced by:  cfeq0  7766  cflim2  7773  cfidm  7785  alephsing  7786  alephreg  8084  pwcfsdom  8085  rankcf  8279
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-en 6750  df-card 7456  df-cf 7458
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