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Theorem gch2 8556
Description: It is sufficient to require that all alephs are GCH-sets to ensure the full generalized continuum hypothesis. (The proof uses the Axiom of Regularity.) (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
gch2  |-  (GCH  =  _V 
<->  ran  aleph  C_ GCH )

Proof of Theorem gch2
StepHypRef Expression
1 ssv 3370 . . 3  |-  ran  aleph  C_  _V
2 sseq2 3372 . . 3  |-  (GCH  =  _V  ->  ( ran  aleph  C_ GCH  <->  ran  aleph  C_  _V ) )
31, 2mpbiri 226 . 2  |-  (GCH  =  _V  ->  ran  aleph  C_ GCH )
4 cardidm 7848 . . . . . . . 8  |-  ( card `  ( card `  x
) )  =  (
card `  x )
5 iscard3 7976 . . . . . . . 8  |-  ( (
card `  ( card `  x ) )  =  ( card `  x
)  <->  ( card `  x
)  e.  ( om  u.  ran  aleph ) )
64, 5mpbi 201 . . . . . . 7  |-  ( card `  x )  e.  ( om  u.  ran  aleph )
7 elun 3490 . . . . . . 7  |-  ( (
card `  x )  e.  ( om  u.  ran  aleph
)  <->  ( ( card `  x )  e.  om  \/  ( card `  x
)  e.  ran  aleph ) )
86, 7mpbi 201 . . . . . 6  |-  ( (
card `  x )  e.  om  \/  ( card `  x )  e.  ran  aleph
)
9 fingch 8500 . . . . . . . . 9  |-  Fin  C_ GCH
10 nnfi 7301 . . . . . . . . 9  |-  ( (
card `  x )  e.  om  ->  ( card `  x )  e.  Fin )
119, 10sseldi 3348 . . . . . . . 8  |-  ( (
card `  x )  e.  om  ->  ( card `  x )  e. GCH )
1211a1i 11 . . . . . . 7  |-  ( ran  aleph 
C_ GCH  ->  ( ( card `  x )  e.  om  ->  ( card `  x
)  e. GCH ) )
13 ssel 3344 . . . . . . 7  |-  ( ran  aleph 
C_ GCH  ->  ( ( card `  x )  e.  ran  aleph  ->  ( card `  x
)  e. GCH ) )
1412, 13jaod 371 . . . . . 6  |-  ( ran  aleph 
C_ GCH  ->  ( ( (
card `  x )  e.  om  \/  ( card `  x )  e.  ran  aleph
)  ->  ( card `  x )  e. GCH )
)
158, 14mpi 17 . . . . 5  |-  ( ran  aleph 
C_ GCH  ->  ( card `  x
)  e. GCH )
16 vex 2961 . . . . . . 7  |-  x  e. 
_V
17 alephon 7952 . . . . . . . . . . 11  |-  ( aleph ` 
suc  x )  e.  On
18 simpr 449 . . . . . . . . . . . 12  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  x  e.  On )
19 simpl 445 . . . . . . . . . . . . 13  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ran  aleph  C_ GCH )
20 alephfnon 7948 . . . . . . . . . . . . . 14  |-  aleph  Fn  On
21 fnfvelrn 5869 . . . . . . . . . . . . . 14  |-  ( (
aleph  Fn  On  /\  x  e.  On )  ->  ( aleph `  x )  e. 
ran  aleph )
2220, 18, 21sylancr 646 . . . . . . . . . . . . 13  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  x )  e. 
ran  aleph )
2319, 22sseldd 3351 . . . . . . . . . . . 12  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  x )  e. GCH )
24 suceloni 4795 . . . . . . . . . . . . . . 15  |-  ( x  e.  On  ->  suc  x  e.  On )
2524adantl 454 . . . . . . . . . . . . . 14  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  suc  x  e.  On )
26 fnfvelrn 5869 . . . . . . . . . . . . . 14  |-  ( (
aleph  Fn  On  /\  suc  x  e.  On )  ->  ( aleph `  suc  x )  e.  ran  aleph )
2720, 25, 26sylancr 646 . . . . . . . . . . . . 13  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  suc  x )  e.  ran  aleph )
2819, 27sseldd 3351 . . . . . . . . . . . 12  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  suc  x )  e. GCH )
29 gchaleph2 8553 . . . . . . . . . . . 12  |-  ( ( x  e.  On  /\  ( aleph `  x )  e. GCH  /\  ( aleph `  suc  x )  e. GCH )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
3018, 23, 28, 29syl3anc 1185 . . . . . . . . . . 11  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
31 isnumi 7835 . . . . . . . . . . 11  |-  ( ( ( aleph `  suc  x )  e.  On  /\  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )  ->  ~P ( aleph `  x )  e.  dom  card )
3217, 30, 31sylancr 646 . . . . . . . . . 10  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ~P ( aleph `  x )  e.  dom  card )
3332ralrimiva 2791 . . . . . . . . 9  |-  ( ran  aleph 
C_ GCH  ->  A. x  e.  On  ~P ( aleph `  x )  e.  dom  card )
34 dfac12 8031 . . . . . . . . 9  |-  (CHOICE  <->  A. x  e.  On  ~P ( aleph `  x )  e.  dom  card )
3533, 34sylibr 205 . . . . . . . 8  |-  ( ran  aleph 
C_ GCH  -> CHOICE
)
36 dfac10 8019 . . . . . . . 8  |-  (CHOICE  <->  dom  card  =  _V )
3735, 36sylib 190 . . . . . . 7  |-  ( ran  aleph 
C_ GCH  ->  dom  card  =  _V )
3816, 37syl5eleqr 2525 . . . . . 6  |-  ( ran  aleph 
C_ GCH  ->  x  e.  dom  card )
39 cardid2 7842 . . . . . 6  |-  ( x  e.  dom  card  ->  (
card `  x )  ~~  x )
40 engch 8505 . . . . . 6  |-  ( (
card `  x )  ~~  x  ->  ( (
card `  x )  e. GCH  <-> 
x  e. GCH ) )
4138, 39, 403syl 19 . . . . 5  |-  ( ran  aleph 
C_ GCH  ->  ( ( card `  x )  e. GCH  <->  x  e. GCH ) )
4215, 41mpbid 203 . . . 4  |-  ( ran  aleph 
C_ GCH  ->  x  e. GCH )
4316a1i 11 . . . 4  |-  ( ran  aleph 
C_ GCH  ->  x  e.  _V )
4442, 432thd 233 . . 3  |-  ( ran  aleph 
C_ GCH  ->  ( x  e. GCH  <->  x  e.  _V ) )
4544eqrdv 2436 . 2  |-  ( ran  aleph 
C_ GCH  -> GCH  =  _V )
463, 45impbii 182 1  |-  (GCH  =  _V 
<->  ran  aleph  C_ GCH )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    u. cun 3320    C_ wss 3322   ~Pcpw 3801   class class class wbr 4214   Oncon0 4583   suc csuc 4585   omcom 4847   dom cdm 4880   ran crn 4881    Fn wfn 5451   ` cfv 5456    ~~ cen 7108   Fincfn 7111   cardccrd 7824   alephcale 7825  CHOICEwac 7998  GCHcgch 8497
This theorem is referenced by:  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-reg 7562  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-r1 7692  df-rank 7693  df-card 7828  df-aleph 7829  df-ac 7999  df-cda 8050  df-fin4 8169  df-gch 8498
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