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Theorem dfac12k 8058
Description: Equivalence of dfac12 8060 and dfac12a 8059, without using Regularity. (Contributed by Mario Carneiro, 21-May-2015.)
Assertion
Ref Expression
dfac12k  |-  ( A. x  e.  On  ~P x  e.  dom  card  <->  A. y  e.  On  ~P ( aleph `  y )  e.  dom  card )
Distinct variable group:    x, y

Proof of Theorem dfac12k
StepHypRef Expression
1 alephon 7981 . . . 4  |-  ( aleph `  y )  e.  On
2 pweq 3826 . . . . . 6  |-  ( x  =  ( aleph `  y
)  ->  ~P x  =  ~P ( aleph `  y
) )
32eleq1d 2508 . . . . 5  |-  ( x  =  ( aleph `  y
)  ->  ( ~P x  e.  dom  card  <->  ~P ( aleph `  y )  e. 
dom  card ) )
43rspcv 3054 . . . 4  |-  ( (
aleph `  y )  e.  On  ->  ( A. x  e.  On  ~P x  e.  dom  card  ->  ~P ( aleph `  y )  e.  dom  card ) )
51, 4ax-mp 5 . . 3  |-  ( A. x  e.  On  ~P x  e.  dom  card  ->  ~P ( aleph `  y )  e.  dom  card )
65ralrimivw 2796 . 2  |-  ( A. x  e.  On  ~P x  e.  dom  card  ->  A. y  e.  On  ~P ( aleph `  y )  e.  dom  card )
7 omelon 7630 . . . . . . 7  |-  om  e.  On
8 cardon 7862 . . . . . . 7  |-  ( card `  x )  e.  On
9 ontri1 4644 . . . . . . 7  |-  ( ( om  e.  On  /\  ( card `  x )  e.  On )  ->  ( om  C_  ( card `  x
)  <->  -.  ( card `  x )  e.  om ) )
107, 8, 9mp2an 655 . . . . . 6  |-  ( om  C_  ( card `  x
)  <->  -.  ( card `  x )  e.  om )
11 cardidm 7877 . . . . . . . 8  |-  ( card `  ( card `  x
) )  =  (
card `  x )
12 cardalephex 8002 . . . . . . . 8  |-  ( om  C_  ( card `  x
)  ->  ( ( card `  ( card `  x
) )  =  (
card `  x )  <->  E. y  e.  On  ( card `  x )  =  ( aleph `  y )
) )
1311, 12mpbii 204 . . . . . . 7  |-  ( om  C_  ( card `  x
)  ->  E. y  e.  On  ( card `  x
)  =  ( aleph `  y ) )
14 r19.29 2852 . . . . . . . . 9  |-  ( ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  /\  E. y  e.  On  ( card `  x
)  =  ( aleph `  y ) )  ->  E. y  e.  On  ( ~P ( aleph `  y
)  e.  dom  card  /\  ( card `  x
)  =  ( aleph `  y ) ) )
15 pweq 3826 . . . . . . . . . . . 12  |-  ( (
card `  x )  =  ( aleph `  y
)  ->  ~P ( card `  x )  =  ~P ( aleph `  y
) )
1615eleq1d 2508 . . . . . . . . . . 11  |-  ( (
card `  x )  =  ( aleph `  y
)  ->  ( ~P ( card `  x )  e.  dom  card  <->  ~P ( aleph `  y
)  e.  dom  card ) )
1716biimparc 475 . . . . . . . . . 10  |-  ( ( ~P ( aleph `  y
)  e.  dom  card  /\  ( card `  x
)  =  ( aleph `  y ) )  ->  ~P ( card `  x
)  e.  dom  card )
1817rexlimivw 2832 . . . . . . . . 9  |-  ( E. y  e.  On  ( ~P ( aleph `  y )  e.  dom  card  /\  ( card `  x )  =  ( aleph `  y )
)  ->  ~P ( card `  x )  e. 
dom  card )
1914, 18syl 16 . . . . . . . 8  |-  ( ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  /\  E. y  e.  On  ( card `  x
)  =  ( aleph `  y ) )  ->  ~P ( card `  x
)  e.  dom  card )
2019ex 425 . . . . . . 7  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ( E. y  e.  On  ( card `  x )  =  ( aleph `  y )  ->  ~P ( card `  x
)  e.  dom  card ) )
2113, 20syl5 31 . . . . . 6  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ( om  C_  ( card `  x
)  ->  ~P ( card `  x )  e. 
dom  card ) )
2210, 21syl5bir 211 . . . . 5  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ( -.  ( card `  x
)  e.  om  ->  ~P ( card `  x
)  e.  dom  card ) )
23 nnfi 7328 . . . . . . 7  |-  ( (
card `  x )  e.  om  ->  ( card `  x )  e.  Fin )
24 pwfi 7431 . . . . . . 7  |-  ( (
card `  x )  e.  Fin  <->  ~P ( card `  x
)  e.  Fin )
2523, 24sylib 190 . . . . . 6  |-  ( (
card `  x )  e.  om  ->  ~P ( card `  x )  e. 
Fin )
26 finnum 7866 . . . . . 6  |-  ( ~P ( card `  x
)  e.  Fin  ->  ~P ( card `  x
)  e.  dom  card )
2725, 26syl 16 . . . . 5  |-  ( (
card `  x )  e.  om  ->  ~P ( card `  x )  e. 
dom  card )
2822, 27pm2.61d2 155 . . . 4  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ~P ( card `  x )  e. 
dom  card )
29 oncardid 7874 . . . . 5  |-  ( x  e.  On  ->  ( card `  x )  ~~  x )
30 pwen 7309 . . . . 5  |-  ( (
card `  x )  ~~  x  ->  ~P ( card `  x )  ~~  ~P x )
31 ennum 7865 . . . . 5  |-  ( ~P ( card `  x
)  ~~  ~P x  ->  ( ~P ( card `  x )  e.  dom  card  <->  ~P x  e.  dom  card ) )
3229, 30, 313syl 19 . . . 4  |-  ( x  e.  On  ->  ( ~P ( card `  x
)  e.  dom  card  <->  ~P x  e.  dom  card )
)
3328, 32syl5ibcom 213 . . 3  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ( x  e.  On  ->  ~P x  e.  dom  card )
)
3433ralrimiv 2794 . 2  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  A. x  e.  On  ~P x  e. 
dom  card )
356, 34impbii 182 1  |-  ( A. x  e.  On  ~P x  e.  dom  card  <->  A. y  e.  On  ~P ( aleph `  y )  e.  dom  card )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727   A.wral 2711   E.wrex 2712    C_ wss 3306   ~Pcpw 3823   class class class wbr 4237   Oncon0 4610   omcom 4874   dom cdm 4907   ` cfv 5483    ~~ cen 7135   Fincfn 7138   cardccrd 7853   alephcale 7854
This theorem is referenced by:  dfac12  8060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-2o 6754  df-oadd 6757  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-oi 7508  df-har 7555  df-card 7857  df-aleph 7858
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