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Theorem dfac12k 8575
Description: Equivalence of dfac12 8577 and dfac12a 8576, 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 8498 . . . 4  |-  ( aleph `  y )  e.  On
2 pweq 3988 . . . . . 6  |-  ( x  =  ( aleph `  y
)  ->  ~P x  =  ~P ( aleph `  y
) )
32eleq1d 2498 . . . . 5  |-  ( x  =  ( aleph `  y
)  ->  ( ~P x  e.  dom  card  <->  ~P ( aleph `  y )  e. 
dom  card ) )
43rspcv 3184 . . . 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 2847 . 2  |-  ( A. x  e.  On  ~P x  e.  dom  card  ->  A. y  e.  On  ~P ( aleph `  y )  e.  dom  card )
7 omelon 8151 . . . . . . 7  |-  om  e.  On
8 cardon 8377 . . . . . . 7  |-  ( card `  x )  e.  On
9 ontri1 5476 . . . . . . 7  |-  ( ( om  e.  On  /\  ( card `  x )  e.  On )  ->  ( om  C_  ( card `  x
)  <->  -.  ( card `  x )  e.  om ) )
107, 8, 9mp2an 676 . . . . . 6  |-  ( om  C_  ( card `  x
)  <->  -.  ( card `  x )  e.  om )
11 cardidm 8392 . . . . . . . 8  |-  ( card `  ( card `  x
) )  =  (
card `  x )
12 cardalephex 8519 . . . . . . . 8  |-  ( om  C_  ( card `  x
)  ->  ( ( card `  ( card `  x
) )  =  (
card `  x )  <->  E. y  e.  On  ( card `  x )  =  ( aleph `  y )
) )
1311, 12mpbii 214 . . . . . . 7  |-  ( om  C_  ( card `  x
)  ->  E. y  e.  On  ( card `  x
)  =  ( aleph `  y ) )
14 r19.29 2970 . . . . . . . . 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 3988 . . . . . . . . . . . 12  |-  ( (
card `  x )  =  ( aleph `  y
)  ->  ~P ( card `  x )  =  ~P ( aleph `  y
) )
1615eleq1d 2498 . . . . . . . . . . 11  |-  ( (
card `  x )  =  ( aleph `  y
)  ->  ( ~P ( card `  x )  e.  dom  card  <->  ~P ( aleph `  y
)  e.  dom  card ) )
1716biimparc 489 . . . . . . . . . 10  |-  ( ( ~P ( aleph `  y
)  e.  dom  card  /\  ( card `  x
)  =  ( aleph `  y ) )  ->  ~P ( card `  x
)  e.  dom  card )
1817rexlimivw 2921 . . . . . . . . 9  |-  ( E. y  e.  On  ( ~P ( aleph `  y )  e.  dom  card  /\  ( card `  x )  =  ( aleph `  y )
)  ->  ~P ( card `  x )  e. 
dom  card )
1914, 18syl 17 . . . . . . . 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 435 . . . . . . 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 33 . . . . . 6  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ( om  C_  ( card `  x
)  ->  ~P ( card `  x )  e. 
dom  card ) )
2210, 21syl5bir 221 . . . . 5  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ( -.  ( card `  x
)  e.  om  ->  ~P ( card `  x
)  e.  dom  card ) )
23 nnfi 7771 . . . . . . 7  |-  ( (
card `  x )  e.  om  ->  ( card `  x )  e.  Fin )
24 pwfi 7875 . . . . . . 7  |-  ( (
card `  x )  e.  Fin  <->  ~P ( card `  x
)  e.  Fin )
2523, 24sylib 199 . . . . . 6  |-  ( (
card `  x )  e.  om  ->  ~P ( card `  x )  e. 
Fin )
26 finnum 8381 . . . . . 6  |-  ( ~P ( card `  x
)  e.  Fin  ->  ~P ( card `  x
)  e.  dom  card )
2725, 26syl 17 . . . . 5  |-  ( (
card `  x )  e.  om  ->  ~P ( card `  x )  e. 
dom  card )
2822, 27pm2.61d2 163 . . . 4  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ~P ( card `  x )  e. 
dom  card )
29 oncardid 8389 . . . . 5  |-  ( x  e.  On  ->  ( card `  x )  ~~  x )
30 pwen 7751 . . . . 5  |-  ( (
card `  x )  ~~  x  ->  ~P ( card `  x )  ~~  ~P x )
31 ennum 8380 . . . . 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 223 . . 3  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  ( x  e.  On  ->  ~P x  e.  dom  card )
)
3433ralrimiv 2844 . 2  |-  ( A. y  e.  On  ~P ( aleph `  y )  e.  dom  card  ->  A. x  e.  On  ~P x  e. 
dom  card )
356, 34impbii 190 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 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783    C_ wss 3442   ~Pcpw 3985   class class class wbr 4426   dom cdm 4854   Oncon0 5442   ` cfv 5601   omcom 6706    ~~ cen 7574   Fincfn 7577   cardccrd 8368   alephcale 8369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-oi 8025  df-har 8073  df-card 8372  df-aleph 8373
This theorem is referenced by:  dfac12  8577
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