MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gchaleph2 Unicode version

Theorem gchaleph2 8298
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 7275 . . 3  |-  (har `  ( aleph `  A )
)  e.  On
2 alephon 7696 . . . . 5  |-  ( aleph `  A )  e.  On
3 onenon 7582 . . . . 5  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
4 harsdom 7628 . . . . 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 7709 . . . . . . 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 6907 . . . . . 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 7695 . . . . . . 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 2358 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph `  suc  A )  e. GCH )  -> 
(har `  ( aleph `  A
) )  e. GCH )
16 gchpwdom 8296 . . . . 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 7664 . . 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 8297 . 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 1623    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   class class class wbr 4023   Oncon0 4392   suc csuc 4394   omcom 4656   dom cdm 4689   ` cfv 5255    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862  harchar 7270   cardccrd 7568   alephcale 7569  GCHcgch 8242
This theorem is referenced by:  gch2  8301  gch3  8302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-oexp 6485  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-wdom 7273  df-cnf 7363  df-card 7572  df-aleph 7573  df-cda 7794  df-fin4 7913  df-gch 8243
  Copyright terms: Public domain W3C validator