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Theorem gchaleph2 8343
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 7320 . . 3  |-  (har `  ( aleph `  A )
)  e.  On
2 alephon 7741 . . . . 5  |-  ( aleph `  A )  e.  On
3 onenon 7627 . . . . 5  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
4 harsdom 7673 . . . . 5  |-  ( (
aleph `  A )  e. 
dom  card  ->  ( aleph `  A )  ~<  (har `  ( aleph `  A )
) )
52, 3, 4mp2b 9 . . . 4  |-  ( aleph `  A )  ~<  (har `  ( aleph `  A )
)
6 simp1 955 . . . . . . 7  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  ->  A  e.  On )
7 alephgeom 7754 . . . . . . 7  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
86, 7sylib 188 . . . . . 6  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  ->  om  C_  ( aleph `  A
) )
9 ssdomg 6950 . . . . . 6  |-  ( (
aleph `  A )  e.  On  ->  ( om  C_  ( aleph `  A )  ->  om  ~<_  ( aleph `  A
) ) )
102, 8, 9mpsyl 59 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  ->  om 
~<_  ( aleph `  A )
)
11 simp2 956 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
( aleph `  A )  e. GCH )
12 alephsuc 7740 . . . . . . 7  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  (har `  ( aleph `  A ) ) )
136, 12syl 15 . . . . . 6  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
( aleph `  suc  A )  =  (har `  ( aleph `  A ) ) )
14 simp3 957 . . . . . 6  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
( aleph `  suc  A )  e. GCH )
1513, 14eqeltrrd 2391 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
(har `  ( aleph `  A
) )  e. GCH )
16 gchpwdom 8341 . . . . 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 1182 . . . 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 202 . . 3  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  ->  ~P ( aleph `  A )  ~<_  (har `  ( aleph `  A
) ) )
19 ondomen 7709 . . 3  |-  ( ( (har `  ( aleph `  A
) )  e.  On  /\ 
~P ( aleph `  A
)  ~<_  (har `  ( aleph `  A ) ) )  ->  ~P ( aleph `  A )  e. 
dom  card )
201, 18, 19sylancr 644 . 2  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  ->  ~P ( aleph `  A )  e.  dom  card )
21 gchaleph 8342 . 2  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( aleph ` 
suc  A )  ~~  ~P ( aleph `  A )
)
2220, 21syld3an3 1227 1  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
( aleph `  suc  A ) 
~~  ~P ( aleph `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1633    e. wcel 1701    C_ wss 3186   ~Pcpw 3659   class class class wbr 4060   Oncon0 4429   suc csuc 4431   omcom 4693   dom cdm 4726   ` cfv 5292    ~~ cen 6903    ~<_ cdom 6904    ~< csdm 6905  harchar 7315   cardccrd 7613   alephcale 7614  GCHcgch 8287
This theorem is referenced by:  gch2  8346  gch3  8347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-seqom 6502  df-1o 6521  df-2o 6522  df-oadd 6525  df-omul 6526  df-oexp 6527  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-oi 7270  df-har 7317  df-wdom 7318  df-cnf 7408  df-card 7617  df-aleph 7618  df-cda 7839  df-fin4 7958  df-gch 8288
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