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Theorem gchaleph2 9051
Description: If  ( aleph `  A ) and  ( aleph `  suc  A ) are GCH-sets, then the successor aleph  ( aleph `  suc  A ) is equinumerous to the powerset of  ( aleph `  A
). (Contributed by Mario Carneiro, 31-May-2015.)
Assertion
Ref Expression
gchaleph2  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
( aleph `  suc  A ) 
~~  ~P ( aleph `  A
) )

Proof of Theorem gchaleph2
StepHypRef Expression
1 harcl 7988 . . 3  |-  (har `  ( aleph `  A )
)  e.  On
2 alephon 8451 . . . . 5  |-  ( aleph `  A )  e.  On
3 onenon 8331 . . . . 5  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
4 harsdom 8377 . . . . 5  |-  ( (
aleph `  A )  e. 
dom  card  ->  ( aleph `  A )  ~<  (har `  ( aleph `  A )
) )
52, 3, 4mp2b 10 . . . 4  |-  ( aleph `  A )  ~<  (har `  ( aleph `  A )
)
6 simp1 996 . . . . . . 7  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  ->  A  e.  On )
7 alephgeom 8464 . . . . . . 7  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
86, 7sylib 196 . . . . . 6  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  ->  om  C_  ( aleph `  A
) )
9 ssdomg 7562 . . . . . 6  |-  ( (
aleph `  A )  e.  On  ->  ( om  C_  ( aleph `  A )  ->  om  ~<_  ( aleph `  A
) ) )
102, 8, 9mpsyl 63 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  ->  om 
~<_  ( aleph `  A )
)
11 simp2 997 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
( aleph `  A )  e. GCH )
12 alephsuc 8450 . . . . . . 7  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  (har `  ( aleph `  A ) ) )
136, 12syl 16 . . . . . 6  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
( aleph `  suc  A )  =  (har `  ( aleph `  A ) ) )
14 simp3 998 . . . . . 6  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
( aleph `  suc  A )  e. GCH )
1513, 14eqeltrrd 2556 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
(har `  ( aleph `  A
) )  e. GCH )
16 gchpwdom 9049 . . . . 5  |-  ( ( om  ~<_  ( aleph `  A
)  /\  ( aleph `  A )  e. GCH  /\  (har `  ( aleph `  A
) )  e. GCH )  ->  ( ( aleph `  A
)  ~<  (har `  ( aleph `  A ) )  <->  ~P ( aleph `  A )  ~<_  (har `  ( aleph `  A
) ) ) )
1710, 11, 15, 16syl3anc 1228 . . . 4  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
( ( aleph `  A
)  ~<  (har `  ( aleph `  A ) )  <->  ~P ( aleph `  A )  ~<_  (har `  ( aleph `  A
) ) ) )
185, 17mpbii 211 . . 3  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  ->  ~P ( aleph `  A )  ~<_  (har `  ( aleph `  A
) ) )
19 ondomen 8419 . . 3  |-  ( ( (har `  ( aleph `  A
) )  e.  On  /\ 
~P ( aleph `  A
)  ~<_  (har `  ( aleph `  A ) ) )  ->  ~P ( aleph `  A )  e. 
dom  card )
201, 18, 19sylancr 663 . 2  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  ->  ~P ( aleph `  A )  e.  dom  card )
21 gchaleph 9050 . 2  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( aleph ` 
suc  A )  ~~  ~P ( aleph `  A )
)
2220, 21syld3an3 1273 1  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
( aleph `  suc  A ) 
~~  ~P ( aleph `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3476   ~Pcpw 4010   class class class wbr 4447   Oncon0 4878   suc csuc 4880   dom cdm 4999   ` cfv 5588   omcom 6685    ~~ cen 7514    ~<_ cdom 7515    ~< csdm 7516  harchar 7983   cardccrd 8317   alephcale 8318  GCHcgch 8999
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 6577  ax-inf2 8059
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6903  df-recs 7043  df-rdg 7077  df-seqom 7114  df-1o 7131  df-2o 7132  df-oadd 7135  df-omul 7136  df-oexp 7137  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7831  df-oi 7936  df-har 7985  df-wdom 7986  df-cnf 8080  df-card 8321  df-aleph 8322  df-cda 8549  df-fin4 8668  df-gch 9000
This theorem is referenced by:  gch2  9054  gch3  9055
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