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Theorem gchaleph 9052
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 8495 . . 3  |-  -.  ~P ( aleph `  A )  ~<  ( aleph `  suc  A )
2 alephon 8453 . . . . 5  |-  ( aleph ` 
suc  A )  e.  On
3 onenon 8333 . . . . 5  |-  ( (
aleph `  suc  A )  e.  On  ->  ( aleph `  suc  A )  e.  dom  card )
42, 3ax-mp 5 . . . 4  |-  ( aleph ` 
suc  A )  e. 
dom  card
5 simp3 999 . . . 4  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ~P ( aleph `  A )  e. 
dom  card )
6 domtri2 8373 . . . 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 663 . . 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 233 . 2  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( aleph ` 
suc  A )  ~<_  ~P ( aleph `  A )
)
9 fvex 5866 . . . . . . 7  |-  ( aleph `  A )  e.  _V
10 simp1 997 . . . . . . . 8  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  A  e.  On )
11 alephgeom 8466 . . . . . . . 8  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
1210, 11sylib 196 . . . . . . 7  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  om  C_  ( aleph `  A ) )
13 ssdomg 7563 . . . . . . 7  |-  ( (
aleph `  A )  e. 
_V  ->  ( om  C_  ( aleph `  A )  ->  om 
~<_  ( aleph `  A )
) )
149, 12, 13mpsyl 63 . . . . . 6  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  om  ~<_  ( aleph `  A ) )
15 domnsym 7645 . . . . . 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 8072 . . . . 5  |-  ( (
aleph `  A )  e. 
Fin 
<->  ( aleph `  A )  ~<  om )
1816, 17sylnibr 305 . . . 4  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  -.  ( aleph `  A )  e. 
Fin )
19 simp2 998 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( aleph `  A )  e. GCH )
20 alephordilem1 8457 . . . . . 6  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
21203ad2ant1 1018 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( aleph `  A )  ~<  ( aleph `  suc  A ) )
22 gchi 9005 . . . . . 6  |-  ( ( ( aleph `  A )  e. GCH  /\  ( aleph `  A
)  ~<  ( aleph `  suc  A )  /\  ( aleph ` 
suc  A )  ~<  ~P ( aleph `  A )
)  ->  ( aleph `  A )  e.  Fin )
23223expia 1199 . . . . 5  |-  ( ( ( aleph `  A )  e. GCH  /\  ( aleph `  A
)  ~<  ( aleph `  suc  A ) )  ->  (
( aleph `  suc  A ) 
~<  ~P ( aleph `  A
)  ->  ( aleph `  A )  e.  Fin ) )
2419, 21, 23syl2anc 661 . . . 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 177 . . 3  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  -.  ( aleph `  suc  A ) 
~<  ~P ( aleph `  A
) )
26 domtri2 8373 . . . 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 662 . . 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 232 . 2  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ~P ( aleph `  A )  ~<_  (
aleph `  suc  A ) )
29 sbth 7639 . 2  |-  ( ( ( aleph `  suc  A )  ~<_  ~P ( aleph `  A
)  /\  ~P ( aleph `  A )  ~<_  (
aleph `  suc  A ) )  ->  ( aleph ` 
suc  A )  ~~  ~P ( aleph `  A )
)
308, 28, 29syl2anc 661 1  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( aleph ` 
suc  A )  ~~  ~P ( aleph `  A )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 974    e. wcel 1804   _Vcvv 3095    C_ wss 3461   ~Pcpw 3997   class class class wbr 4437   Oncon0 4868   suc csuc 4870   dom cdm 4989   ` cfv 5578   omcom 6685    ~~ cen 7515    ~<_ cdom 7516    ~< csdm 7517   Fincfn 7518   cardccrd 8319   alephcale 8320  GCHcgch 9001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-om 6686  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-oi 7938  df-har 7987  df-card 8323  df-aleph 8324  df-gch 9002
This theorem is referenced by:  gchaleph2  9053
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