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Theorem gch2 9042
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 3509 . . 3  |-  ran  aleph  C_  _V
2 sseq2 3511 . . 3  |-  (GCH  =  _V  ->  ( ran  aleph  C_ GCH  <->  ran  aleph  C_  _V ) )
31, 2mpbiri 233 . 2  |-  (GCH  =  _V  ->  ran  aleph  C_ GCH )
4 cardidm 8331 . . . . . . . 8  |-  ( card `  ( card `  x
) )  =  (
card `  x )
5 iscard3 8465 . . . . . . . 8  |-  ( (
card `  ( card `  x ) )  =  ( card `  x
)  <->  ( card `  x
)  e.  ( om  u.  ran  aleph ) )
64, 5mpbi 208 . . . . . . 7  |-  ( card `  x )  e.  ( om  u.  ran  aleph )
7 elun 3631 . . . . . . 7  |-  ( (
card `  x )  e.  ( om  u.  ran  aleph
)  <->  ( ( card `  x )  e.  om  \/  ( card `  x
)  e.  ran  aleph ) )
86, 7mpbi 208 . . . . . 6  |-  ( (
card `  x )  e.  om  \/  ( card `  x )  e.  ran  aleph
)
9 fingch 8990 . . . . . . . . 9  |-  Fin  C_ GCH
10 nnfi 7703 . . . . . . . . 9  |-  ( (
card `  x )  e.  om  ->  ( card `  x )  e.  Fin )
119, 10sseldi 3487 . . . . . . . 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 3483 . . . . . . 7  |-  ( ran  aleph 
C_ GCH  ->  ( ( card `  x )  e.  ran  aleph  ->  ( card `  x
)  e. GCH ) )
1412, 13jaod 378 . . . . . 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 3109 . . . . . . 7  |-  x  e. 
_V
17 alephon 8441 . . . . . . . . . . 11  |-  ( aleph ` 
suc  x )  e.  On
18 simpr 459 . . . . . . . . . . . 12  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  x  e.  On )
19 simpl 455 . . . . . . . . . . . . 13  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ran  aleph  C_ GCH )
20 alephfnon 8437 . . . . . . . . . . . . . 14  |-  aleph  Fn  On
21 fnfvelrn 6004 . . . . . . . . . . . . . 14  |-  ( (
aleph  Fn  On  /\  x  e.  On )  ->  ( aleph `  x )  e. 
ran  aleph )
2220, 18, 21sylancr 661 . . . . . . . . . . . . 13  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  x )  e. 
ran  aleph )
2319, 22sseldd 3490 . . . . . . . . . . . 12  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  x )  e. GCH )
24 suceloni 6621 . . . . . . . . . . . . . . 15  |-  ( x  e.  On  ->  suc  x  e.  On )
2524adantl 464 . . . . . . . . . . . . . 14  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  suc  x  e.  On )
26 fnfvelrn 6004 . . . . . . . . . . . . . 14  |-  ( (
aleph  Fn  On  /\  suc  x  e.  On )  ->  ( aleph `  suc  x )  e.  ran  aleph )
2720, 25, 26sylancr 661 . . . . . . . . . . . . 13  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  suc  x )  e.  ran  aleph )
2819, 27sseldd 3490 . . . . . . . . . . . 12  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  suc  x )  e. GCH )
29 gchaleph2 9039 . . . . . . . . . . . 12  |-  ( ( x  e.  On  /\  ( aleph `  x )  e. GCH  /\  ( aleph `  suc  x )  e. GCH )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
3018, 23, 28, 29syl3anc 1226 . . . . . . . . . . 11  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
31 isnumi 8318 . . . . . . . . . . 11  |-  ( ( ( aleph `  suc  x )  e.  On  /\  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )  ->  ~P ( aleph `  x )  e.  dom  card )
3217, 30, 31sylancr 661 . . . . . . . . . 10  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ~P ( aleph `  x )  e.  dom  card )
3332ralrimiva 2868 . . . . . . . . 9  |-  ( ran  aleph 
C_ GCH  ->  A. x  e.  On  ~P ( aleph `  x )  e.  dom  card )
34 dfac12 8520 . . . . . . . . 9  |-  (CHOICE  <->  A. x  e.  On  ~P ( aleph `  x )  e.  dom  card )
3533, 34sylibr 212 . . . . . . . 8  |-  ( ran  aleph 
C_ GCH  -> CHOICE
)
36 dfac10 8508 . . . . . . . 8  |-  (CHOICE  <->  dom  card  =  _V )
3735, 36sylib 196 . . . . . . 7  |-  ( ran  aleph 
C_ GCH  ->  dom  card  =  _V )
3816, 37syl5eleqr 2549 . . . . . 6  |-  ( ran  aleph 
C_ GCH  ->  x  e.  dom  card )
39 cardid2 8325 . . . . . 6  |-  ( x  e.  dom  card  ->  (
card `  x )  ~~  x )
40 engch 8995 . . . . . 6  |-  ( (
card `  x )  ~~  x  ->  ( (
card `  x )  e. GCH  <-> 
x  e. GCH ) )
4138, 39, 403syl 20 . . . . 5  |-  ( ran  aleph 
C_ GCH  ->  ( ( card `  x )  e. GCH  <->  x  e. GCH ) )
4215, 41mpbid 210 . . . 4  |-  ( ran  aleph 
C_ GCH  ->  x  e. GCH )
4316a1i 11 . . . 4  |-  ( ran  aleph 
C_ GCH  ->  x  e.  _V )
4442, 432thd 240 . . 3  |-  ( ran  aleph 
C_ GCH  ->  ( x  e. GCH  <->  x  e.  _V ) )
4544eqrdv 2451 . 2  |-  ( ran  aleph 
C_ GCH  -> GCH  =  _V )
463, 45impbii 188 1  |-  (GCH  =  _V 
<->  ran  aleph  C_ GCH )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106    u. cun 3459    C_ wss 3461   ~Pcpw 3999   class class class wbr 4439   Oncon0 4867   suc csuc 4869   dom cdm 4988   ran crn 4989    Fn wfn 5565   ` cfv 5570   omcom 6673    ~~ cen 7506   Fincfn 7509   cardccrd 8307   alephcale 8308  CHOICEwac 8487  GCHcgch 8987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-reg 8010  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-seqom 7105  df-1o 7122  df-2o 7123  df-oadd 7126  df-omul 7127  df-oexp 7128  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-har 7976  df-wdom 7977  df-cnf 8070  df-r1 8173  df-rank 8174  df-card 8311  df-aleph 8312  df-ac 8488  df-cda 8539  df-fin4 8658  df-gch 8988
This theorem is referenced by:  gch3  9043
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