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Theorem gchaleph2 9048
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 7985 . . 3  |-  (har `  ( aleph `  A )
)  e.  On
2 alephon 8448 . . . . 5  |-  ( aleph `  A )  e.  On
3 onenon 8328 . . . . 5  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
4 harsdom 8374 . . . . 5  |-  ( (
aleph `  A )  e. 
dom  card  ->  ( aleph `  A )  ~<  (har `  ( aleph `  A )
) )
52, 3, 4mp2b 10 . . . 4  |-  ( aleph `  A )  ~<  (har `  ( aleph `  A )
)
6 simp1 995 . . . . . . 7  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  ->  A  e.  On )
7 alephgeom 8461 . . . . . . 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 7559 . . . . . 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 996 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
( aleph `  A )  e. GCH )
12 alephsuc 8447 . . . . . . 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 997 . . . . . 6  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
( aleph `  suc  A )  e. GCH )
1513, 14eqeltrrd 2530 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
(har `  ( aleph `  A
) )  e. GCH )
16 gchpwdom 9046 . . . . 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 1227 . . . 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 8416 . . 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 9047 . 2  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( aleph ` 
suc  A )  ~~  ~P ( aleph `  A )
)
2220, 21syld3an3 1272 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 972    = wceq 1381    e. wcel 1802    C_ wss 3458   ~Pcpw 3993   class class class wbr 4433   Oncon0 4864   suc csuc 4866   dom cdm 4985   ` cfv 5574   omcom 6681    ~~ cen 7511    ~<_ cdom 7512    ~< csdm 7513  harchar 7980   cardccrd 8314   alephcale 8315  GCHcgch 8996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-seqom 7111  df-1o 7128  df-2o 7129  df-oadd 7132  df-omul 7133  df-oexp 7134  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-oi 7933  df-har 7982  df-wdom 7983  df-cnf 8077  df-card 8318  df-aleph 8319  df-cda 8546  df-fin4 8665  df-gch 8997
This theorem is referenced by:  gch2  9051  gch3  9052
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