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Theorem dfac12k 8527
Description: Equivalence of dfac12 8529 and dfac12a 8528, 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 8450 . . . 4  |-  ( aleph `  y )  e.  On
2 pweq 4013 . . . . . 6  |-  ( x  =  ( aleph `  y
)  ->  ~P x  =  ~P ( aleph `  y
) )
32eleq1d 2536 . . . . 5  |-  ( x  =  ( aleph `  y
)  ->  ( ~P x  e.  dom  card  <->  ~P ( aleph `  y )  e. 
dom  card ) )
43rspcv 3210 . . . 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 2879 . 2  |-  ( A. x  e.  On  ~P x  e.  dom  card  ->  A. y  e.  On  ~P ( aleph `  y )  e.  dom  card )
7 omelon 8063 . . . . . . 7  |-  om  e.  On
8 cardon 8325 . . . . . . 7  |-  ( card `  x )  e.  On
9 ontri1 4912 . . . . . . 7  |-  ( ( om  e.  On  /\  ( card `  x )  e.  On )  ->  ( om  C_  ( card `  x
)  <->  -.  ( card `  x )  e.  om ) )
107, 8, 9mp2an 672 . . . . . 6  |-  ( om  C_  ( card `  x
)  <->  -.  ( card `  x )  e.  om )
11 cardidm 8340 . . . . . . . 8  |-  ( card `  ( card `  x
) )  =  (
card `  x )
12 cardalephex 8471 . . . . . . . 8  |-  ( om  C_  ( card `  x
)  ->  ( ( card `  ( card `  x
) )  =  (
card `  x )  <->  E. y  e.  On  ( card `  x )  =  ( aleph `  y )
) )
1311, 12mpbii 211 . . . . . . 7  |-  ( om  C_  ( card `  x
)  ->  E. y  e.  On  ( card `  x
)  =  ( aleph `  y ) )
14 r19.29 2997 . . . . . . . . 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 4013 . . . . . . . . . . . 12  |-  ( (
card `  x )  =  ( aleph `  y
)  ->  ~P ( card `  x )  =  ~P ( aleph `  y
) )
1615eleq1d 2536 . . . . . . . . . . 11  |-  ( (
card `  x )  =  ( aleph `  y
)  ->  ( ~P ( card `  x )  e.  dom  card  <->  ~P ( aleph `  y
)  e.  dom  card ) )
1716biimparc 487 . . . . . . . . . 10  |-  ( ( ~P ( aleph `  y
)  e.  dom  card  /\  ( card `  x
)  =  ( aleph `  y ) )  ->  ~P ( card `  x
)  e.  dom  card )
1817rexlimivw 2952 . . . . . . . . 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 434 . . . . . . 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 32 . . . . . 6  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ( om  C_  ( card `  x
)  ->  ~P ( card `  x )  e. 
dom  card ) )
2210, 21syl5bir 218 . . . . 5  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ( -.  ( card `  x
)  e.  om  ->  ~P ( card `  x
)  e.  dom  card ) )
23 nnfi 7710 . . . . . . 7  |-  ( (
card `  x )  e.  om  ->  ( card `  x )  e.  Fin )
24 pwfi 7815 . . . . . . 7  |-  ( (
card `  x )  e.  Fin  <->  ~P ( card `  x
)  e.  Fin )
2523, 24sylib 196 . . . . . 6  |-  ( (
card `  x )  e.  om  ->  ~P ( card `  x )  e. 
Fin )
26 finnum 8329 . . . . . 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 160 . . . 4  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ~P ( card `  x )  e. 
dom  card )
29 oncardid 8337 . . . . 5  |-  ( x  e.  On  ->  ( card `  x )  ~~  x )
30 pwen 7690 . . . . 5  |-  ( (
card `  x )  ~~  x  ->  ~P ( card `  x )  ~~  ~P x )
31 ennum 8328 . . . . 5  |-  ( ~P ( card `  x
)  ~~  ~P x  ->  ( ~P ( card `  x )  e.  dom  card  <->  ~P x  e.  dom  card ) )
3229, 30, 313syl 20 . . . 4  |-  ( x  e.  On  ->  ( ~P ( card `  x
)  e.  dom  card  <->  ~P x  e.  dom  card )
)
3328, 32syl5ibcom 220 . . 3  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ( x  e.  On  ->  ~P x  e.  dom  card )
)
3433ralrimiv 2876 . 2  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  A. x  e.  On  ~P x  e. 
dom  card )
356, 34impbii 188 1  |-  ( A. x  e.  On  ~P x  e.  dom  card  <->  A. y  e.  On  ~P ( aleph `  y )  e.  dom  card )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   ~Pcpw 4010   class class class wbr 4447   Oncon0 4878   dom cdm 4999   ` cfv 5588   omcom 6684    ~~ cen 7513   Fincfn 7516   cardccrd 8316   alephcale 8317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-oi 7935  df-har 7984  df-card 8320  df-aleph 8321
This theorem is referenced by:  dfac12  8529
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