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Theorem gchaleph 8550
Description: If  ( aleph `  A ) is a GCH-set and its powerset is well-orderable, then the successor aleph  ( aleph `  suc  A ) is equinumerous to the powerset of  ( aleph `  A
). (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
gchaleph  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( aleph ` 
suc  A )  ~~  ~P ( aleph `  A )
)

Proof of Theorem gchaleph
StepHypRef Expression
1 alephsucpw2 7992 . . 3  |-  -.  ~P ( aleph `  A )  ~<  ( aleph `  suc  A )
2 alephon 7950 . . . . 5  |-  ( aleph ` 
suc  A )  e.  On
3 onenon 7836 . . . . 5  |-  ( (
aleph `  suc  A )  e.  On  ->  ( aleph `  suc  A )  e.  dom  card )
42, 3ax-mp 8 . . . 4  |-  ( aleph ` 
suc  A )  e. 
dom  card
5 simp3 959 . . . 4  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ~P ( aleph `  A )  e. 
dom  card )
6 domtri2 7876 . . . 4  |-  ( ( ( aleph `  suc  A )  e.  dom  card  /\  ~P ( aleph `  A )  e.  dom  card )  ->  (
( aleph `  suc  A )  ~<_  ~P ( aleph `  A
)  <->  -.  ~P ( aleph `  A )  ~< 
( aleph `  suc  A ) ) )
74, 5, 6sylancr 645 . . 3  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( ( aleph `  suc  A )  ~<_  ~P ( aleph `  A
)  <->  -.  ~P ( aleph `  A )  ~< 
( aleph `  suc  A ) ) )
81, 7mpbiri 225 . 2  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( aleph ` 
suc  A )  ~<_  ~P ( aleph `  A )
)
9 fvex 5742 . . . . . . 7  |-  ( aleph `  A )  e.  _V
10 simp1 957 . . . . . . . 8  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  A  e.  On )
11 alephgeom 7963 . . . . . . . 8  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
1210, 11sylib 189 . . . . . . 7  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  om  C_  ( aleph `  A ) )
13 ssdomg 7153 . . . . . . 7  |-  ( (
aleph `  A )  e. 
_V  ->  ( om  C_  ( aleph `  A )  ->  om 
~<_  ( aleph `  A )
) )
149, 12, 13mpsyl 61 . . . . . 6  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  om  ~<_  ( aleph `  A ) )
15 domnsym 7233 . . . . . 6  |-  ( om  ~<_  ( aleph `  A )  ->  -.  ( aleph `  A
)  ~<  om )
1614, 15syl 16 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  -.  ( aleph `  A )  ~<  om )
17 isfinite 7607 . . . . 5  |-  ( (
aleph `  A )  e. 
Fin 
<->  ( aleph `  A )  ~<  om )
1816, 17sylnibr 297 . . . 4  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  -.  ( aleph `  A )  e. 
Fin )
19 simp2 958 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( aleph `  A )  e. GCH )
20 alephordilem1 7954 . . . . . 6  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
21203ad2ant1 978 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( aleph `  A )  ~<  ( aleph `  suc  A ) )
22 gchi 8499 . . . . . 6  |-  ( ( ( aleph `  A )  e. GCH  /\  ( aleph `  A
)  ~<  ( aleph `  suc  A )  /\  ( aleph ` 
suc  A )  ~<  ~P ( aleph `  A )
)  ->  ( aleph `  A )  e.  Fin )
23223expia 1155 . . . . 5  |-  ( ( ( aleph `  A )  e. GCH  /\  ( aleph `  A
)  ~<  ( aleph `  suc  A ) )  ->  (
( aleph `  suc  A ) 
~<  ~P ( aleph `  A
)  ->  ( aleph `  A )  e.  Fin ) )
2419, 21, 23syl2anc 643 . . . 4  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( ( aleph `  suc  A ) 
~<  ~P ( aleph `  A
)  ->  ( aleph `  A )  e.  Fin ) )
2518, 24mtod 170 . . 3  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  -.  ( aleph `  suc  A ) 
~<  ~P ( aleph `  A
) )
26 domtri2 7876 . . . 4  |-  ( ( ~P ( aleph `  A
)  e.  dom  card  /\  ( aleph `  suc  A )  e.  dom  card )  ->  ( ~P ( aleph `  A )  ~<_  ( aleph ` 
suc  A )  <->  -.  ( aleph `  suc  A ) 
~<  ~P ( aleph `  A
) ) )
275, 4, 26sylancl 644 . . 3  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( ~P ( aleph `  A )  ~<_  ( aleph `  suc  A )  <->  -.  ( aleph `  suc  A ) 
~<  ~P ( aleph `  A
) ) )
2825, 27mpbird 224 . 2  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ~P ( aleph `  A )  ~<_  (
aleph `  suc  A ) )
29 sbth 7227 . 2  |-  ( ( ( aleph `  suc  A )  ~<_  ~P ( aleph `  A
)  /\  ~P ( aleph `  A )  ~<_  (
aleph `  suc  A ) )  ->  ( aleph ` 
suc  A )  ~~  ~P ( aleph `  A )
)
308, 28, 29syl2anc 643 1  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( aleph ` 
suc  A )  ~~  ~P ( aleph `  A )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ w3a 936    e. wcel 1725   _Vcvv 2956    C_ wss 3320   ~Pcpw 3799   class class class wbr 4212   Oncon0 4581   suc csuc 4583   omcom 4845   dom cdm 4878   ` cfv 5454    ~~ cen 7106    ~<_ cdom 7107    ~< csdm 7108   Fincfn 7109   cardccrd 7822   alephcale 7823  GCHcgch 8495
This theorem is referenced by:  gchaleph2  8551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-har 7526  df-card 7826  df-aleph 7827  df-gch 8496
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