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Theorem gchaleph2 8553
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 7531 . . 3  |-  (har `  ( aleph `  A )
)  e.  On
2 alephon 7952 . . . . 5  |-  ( aleph `  A )  e.  On
3 onenon 7838 . . . . 5  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
4 harsdom 7884 . . . . 5  |-  ( (
aleph `  A )  e. 
dom  card  ->  ( aleph `  A )  ~<  (har `  ( aleph `  A )
) )
52, 3, 4mp2b 10 . . . 4  |-  ( aleph `  A )  ~<  (har `  ( aleph `  A )
)
6 simp1 958 . . . . . . 7  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  ->  A  e.  On )
7 alephgeom 7965 . . . . . . 7  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
86, 7sylib 190 . . . . . 6  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  ->  om  C_  ( aleph `  A
) )
9 ssdomg 7155 . . . . . 6  |-  ( (
aleph `  A )  e.  On  ->  ( om  C_  ( aleph `  A )  ->  om  ~<_  ( aleph `  A
) ) )
102, 8, 9mpsyl 62 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  ->  om 
~<_  ( aleph `  A )
)
11 simp2 959 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
( aleph `  A )  e. GCH )
12 alephsuc 7951 . . . . . . 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 960 . . . . . 6  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
( aleph `  suc  A )  e. GCH )
1513, 14eqeltrrd 2513 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
(har `  ( aleph `  A
) )  e. GCH )
16 gchpwdom 8551 . . . . 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 1185 . . . 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 204 . . 3  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  ->  ~P ( aleph `  A )  ~<_  (har `  ( aleph `  A
) ) )
19 ondomen 7920 . . 3  |-  ( ( (har `  ( aleph `  A
) )  e.  On  /\ 
~P ( aleph `  A
)  ~<_  (har `  ( aleph `  A ) ) )  ->  ~P ( aleph `  A )  e. 
dom  card )
201, 18, 19sylancr 646 . 2  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  ->  ~P ( aleph `  A )  e.  dom  card )
21 gchaleph 8552 . 2  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( aleph ` 
suc  A )  ~~  ~P ( aleph `  A )
)
2220, 21syld3an3 1230 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 178    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3322   ~Pcpw 3801   class class class wbr 4214   Oncon0 4583   suc csuc 4585   omcom 4847   dom cdm 4880   ` cfv 5456    ~~ cen 7108    ~<_ cdom 7109    ~< csdm 7110  harchar 7526   cardccrd 7824   alephcale 7825  GCHcgch 8497
This theorem is referenced by:  gch2  8556  gch3  8557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-fal 1330  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-seqom 6707  df-1o 6726  df-2o 6727  df-oadd 6730  df-omul 6731  df-oexp 6732  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-oi 7481  df-har 7528  df-wdom 7529  df-cnf 7619  df-card 7828  df-aleph 7829  df-cda 8050  df-fin4 8169  df-gch 8498
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