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Theorem dfac12k 7773
Description: Equivalence of dfac12 7775 and dfac12a 7774, 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 7696 . . . 4  |-  ( aleph `  y )  e.  On
2 pweq 3628 . . . . . 6  |-  ( x  =  ( aleph `  y
)  ->  ~P x  =  ~P ( aleph `  y
) )
32eleq1d 2349 . . . . 5  |-  ( x  =  ( aleph `  y
)  ->  ( ~P x  e.  dom  card  <->  ~P ( aleph `  y )  e. 
dom  card ) )
43rspcv 2880 . . . 4  |-  ( (
aleph `  y )  e.  On  ->  ( A. x  e.  On  ~P x  e.  dom  card  ->  ~P ( aleph `  y )  e.  dom  card ) )
51, 4ax-mp 8 . . 3  |-  ( A. x  e.  On  ~P x  e.  dom  card  ->  ~P ( aleph `  y )  e.  dom  card )
65ralrimivw 2627 . 2  |-  ( A. x  e.  On  ~P x  e.  dom  card  ->  A. y  e.  On  ~P ( aleph `  y )  e.  dom  card )
7 omelon 7347 . . . . . . 7  |-  om  e.  On
8 cardon 7577 . . . . . . 7  |-  ( card `  x )  e.  On
9 ontri1 4426 . . . . . . 7  |-  ( ( om  e.  On  /\  ( card `  x )  e.  On )  ->  ( om  C_  ( card `  x
)  <->  -.  ( card `  x )  e.  om ) )
107, 8, 9mp2an 653 . . . . . 6  |-  ( om  C_  ( card `  x
)  <->  -.  ( card `  x )  e.  om )
11 cardidm 7592 . . . . . . . 8  |-  ( card `  ( card `  x
) )  =  (
card `  x )
12 cardalephex 7717 . . . . . . . 8  |-  ( om  C_  ( card `  x
)  ->  ( ( card `  ( card `  x
) )  =  (
card `  x )  <->  E. y  e.  On  ( card `  x )  =  ( aleph `  y )
) )
1311, 12mpbii 202 . . . . . . 7  |-  ( om  C_  ( card `  x
)  ->  E. y  e.  On  ( card `  x
)  =  ( aleph `  y ) )
14 r19.29 2683 . . . . . . . . 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 3628 . . . . . . . . . . . 12  |-  ( (
card `  x )  =  ( aleph `  y
)  ->  ~P ( card `  x )  =  ~P ( aleph `  y
) )
1615eleq1d 2349 . . . . . . . . . . 11  |-  ( (
card `  x )  =  ( aleph `  y
)  ->  ( ~P ( card `  x )  e.  dom  card  <->  ~P ( aleph `  y
)  e.  dom  card ) )
1716biimparc 473 . . . . . . . . . 10  |-  ( ( ~P ( aleph `  y
)  e.  dom  card  /\  ( card `  x
)  =  ( aleph `  y ) )  ->  ~P ( card `  x
)  e.  dom  card )
1817rexlimivw 2663 . . . . . . . . 9  |-  ( E. y  e.  On  ( ~P ( aleph `  y )  e.  dom  card  /\  ( card `  x )  =  ( aleph `  y )
)  ->  ~P ( card `  x )  e. 
dom  card )
1914, 18syl 15 . . . . . . . 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 423 . . . . . . 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 28 . . . . . 6  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ( om  C_  ( card `  x
)  ->  ~P ( card `  x )  e. 
dom  card ) )
2210, 21syl5bir 209 . . . . 5  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ( -.  ( card `  x
)  e.  om  ->  ~P ( card `  x
)  e.  dom  card ) )
23 nnfi 7053 . . . . . . 7  |-  ( (
card `  x )  e.  om  ->  ( card `  x )  e.  Fin )
24 pwfi 7151 . . . . . . 7  |-  ( (
card `  x )  e.  Fin  <->  ~P ( card `  x
)  e.  Fin )
2523, 24sylib 188 . . . . . 6  |-  ( (
card `  x )  e.  om  ->  ~P ( card `  x )  e. 
Fin )
26 finnum 7581 . . . . . 6  |-  ( ~P ( card `  x
)  e.  Fin  ->  ~P ( card `  x
)  e.  dom  card )
2725, 26syl 15 . . . . 5  |-  ( (
card `  x )  e.  om  ->  ~P ( card `  x )  e. 
dom  card )
2822, 27pm2.61d2 152 . . . 4  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ~P ( card `  x )  e. 
dom  card )
29 oncardid 7589 . . . . 5  |-  ( x  e.  On  ->  ( card `  x )  ~~  x )
30 pwen 7034 . . . . 5  |-  ( (
card `  x )  ~~  x  ->  ~P ( card `  x )  ~~  ~P x )
31 ennum 7580 . . . . 5  |-  ( ~P ( card `  x
)  ~~  ~P x  ->  ( ~P ( card `  x )  e.  dom  card  <->  ~P x  e.  dom  card ) )
3229, 30, 313syl 18 . . . 4  |-  ( x  e.  On  ->  ( ~P ( card `  x
)  e.  dom  card  <->  ~P x  e.  dom  card )
)
3328, 32syl5ibcom 211 . . 3  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ( x  e.  On  ->  ~P x  e.  dom  card )
)
3433ralrimiv 2625 . 2  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  A. x  e.  On  ~P x  e. 
dom  card )
356, 34impbii 180 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   ~Pcpw 3625   class class class wbr 4023   Oncon0 4392   omcom 4656   dom cdm 4689   ` cfv 5255    ~~ cen 6860   Fincfn 6863   cardccrd 7568   alephcale 7569
This theorem is referenced by:  dfac12  7775
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-card 7572  df-aleph 7573
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