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Theorem dfac12k 7789
Description: Equivalence of dfac12 7791 and dfac12a 7790, 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 7712 . . . 4  |-  ( aleph `  y )  e.  On
2 pweq 3641 . . . . . 6  |-  ( x  =  ( aleph `  y
)  ->  ~P x  =  ~P ( aleph `  y
) )
32eleq1d 2362 . . . . 5  |-  ( x  =  ( aleph `  y
)  ->  ( ~P x  e.  dom  card  <->  ~P ( aleph `  y )  e. 
dom  card ) )
43rspcv 2893 . . . 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 2640 . 2  |-  ( A. x  e.  On  ~P x  e.  dom  card  ->  A. y  e.  On  ~P ( aleph `  y )  e.  dom  card )
7 omelon 7363 . . . . . . 7  |-  om  e.  On
8 cardon 7593 . . . . . . 7  |-  ( card `  x )  e.  On
9 ontri1 4442 . . . . . . 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 7608 . . . . . . . 8  |-  ( card `  ( card `  x
) )  =  (
card `  x )
12 cardalephex 7733 . . . . . . . 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 2696 . . . . . . . . 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 3641 . . . . . . . . . . . 12  |-  ( (
card `  x )  =  ( aleph `  y
)  ->  ~P ( card `  x )  =  ~P ( aleph `  y
) )
1615eleq1d 2362 . . . . . . . . . . 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 2676 . . . . . . . . 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 7069 . . . . . . 7  |-  ( (
card `  x )  e.  om  ->  ( card `  x )  e.  Fin )
24 pwfi 7167 . . . . . . 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 7597 . . . . . 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 7605 . . . . 5  |-  ( x  e.  On  ->  ( card `  x )  ~~  x )
30 pwen 7050 . . . . 5  |-  ( (
card `  x )  ~~  x  ->  ~P ( card `  x )  ~~  ~P x )
31 ennum 7596 . . . . 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 2638 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   ~Pcpw 3638   class class class wbr 4039   Oncon0 4408   omcom 4672   dom cdm 4705   ` cfv 5271    ~~ cen 6876   Fincfn 6879   cardccrd 7584   alephcale 7585
This theorem is referenced by:  dfac12  7791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-card 7588  df-aleph 7589
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