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Theorem gchaleph 8313
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 7754 . . 3  |-  -.  ~P ( aleph `  A )  ~<  ( aleph `  suc  A )
2 alephon 7712 . . . . 5  |-  ( aleph ` 
suc  A )  e.  On
3 onenon 7598 . . . . 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 957 . . . 4  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ~P ( aleph `  A )  e. 
dom  card )
6 domtri2 7638 . . . 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 644 . . 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 224 . 2  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( aleph ` 
suc  A )  ~<_  ~P ( aleph `  A )
)
9 fvex 5555 . . . . . . 7  |-  ( aleph `  A )  e.  _V
10 simp1 955 . . . . . . . 8  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  A  e.  On )
11 alephgeom 7725 . . . . . . . 8  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
1210, 11sylib 188 . . . . . . 7  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  om  C_  ( aleph `  A ) )
13 ssdomg 6923 . . . . . . 7  |-  ( (
aleph `  A )  e. 
_V  ->  ( om  C_  ( aleph `  A )  ->  om 
~<_  ( aleph `  A )
) )
149, 12, 13mpsyl 59 . . . . . 6  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  om  ~<_  ( aleph `  A ) )
15 domnsym 7003 . . . . . 6  |-  ( om  ~<_  ( aleph `  A )  ->  -.  ( aleph `  A
)  ~<  om )
1614, 15syl 15 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  -.  ( aleph `  A )  ~<  om )
17 isfinite 7369 . . . . 5  |-  ( (
aleph `  A )  e. 
Fin 
<->  ( aleph `  A )  ~<  om )
1816, 17sylnibr 296 . . . 4  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  -.  ( aleph `  A )  e. 
Fin )
19 simp2 956 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( aleph `  A )  e. GCH )
20 alephordilem1 7716 . . . . . 6  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
21203ad2ant1 976 . . . . 5  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ( aleph `  A )  ~<  ( aleph `  suc  A ) )
22 gchi 8262 . . . . . 6  |-  ( ( ( aleph `  A )  e. GCH  /\  ( aleph `  A
)  ~<  ( aleph `  suc  A )  /\  ( aleph ` 
suc  A )  ~<  ~P ( aleph `  A )
)  ->  ( aleph `  A )  e.  Fin )
23223expia 1153 . . . . 5  |-  ( ( ( aleph `  A )  e. GCH  /\  ( aleph `  A
)  ~<  ( aleph `  suc  A ) )  ->  (
( aleph `  suc  A ) 
~<  ~P ( aleph `  A
)  ->  ( aleph `  A )  e.  Fin ) )
2419, 21, 23syl2anc 642 . . . 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 168 . . 3  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  -.  ( aleph `  suc  A ) 
~<  ~P ( aleph `  A
) )
26 domtri2 7638 . . . 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 643 . . 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 223 . 2  |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card )  ->  ~P ( aleph `  A )  ~<_  (
aleph `  suc  A ) )
29 sbth 6997 . 2  |-  ( ( ( aleph `  suc  A )  ~<_  ~P ( aleph `  A
)  /\  ~P ( aleph `  A )  ~<_  (
aleph `  suc  A ) )  ->  ( aleph ` 
suc  A )  ~~  ~P ( aleph `  A )
)
308, 28, 29syl2anc 642 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 176    /\ w3a 934    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   class class class wbr 4039   Oncon0 4408   suc csuc 4410   omcom 4672   dom cdm 4705   ` cfv 5271    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878   Fincfn 6879   cardccrd 7584   alephcale 7585  GCHcgch 8258
This theorem is referenced by:  gchaleph2  8314
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-card 7588  df-aleph 7589  df-gch 8259
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