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Theorem alephom 8207
Description: From canth2 7014, we know that  (
aleph `  0 )  < 
( 2 ^ om ), but we cannot prove that  ( 2 ^ om )  =  ( aleph `  1 ) (this is the Continuum Hypothesis), nor can we prove that it is less than any bound whatsoever (i.e. the statement  ( aleph `  A )  <  ( 2 ^ om ) is consistent for any ordinal  A). However, we can prove that  ( 2 ^ om ) is not equal to  ( aleph `  om ), nor  ( aleph `  ( aleph `  om ) ), on cofinality grounds, because by Konig's Theorem konigth 8191 (in the form of cfpwsdom 8206), 
( 2 ^ om ) has uncountable cofinality, which eliminates limit alephs like 
( aleph `  om ). (The first limit aleph that is not eliminated is  (
aleph `  ( aleph `  1
) ), which has cofinality  ( aleph `  1 ).) (Contributed by Mario Carneiro, 21-Mar-2013.)
Assertion
Ref Expression
alephom  |-  ( card `  ( 2o  ^m  om ) )  =/=  ( aleph `  om )

Proof of Theorem alephom
StepHypRef Expression
1 sdomirr 6998 . 2  |-  -.  om  ~<  om
2 2onn 6638 . . . . . 6  |-  2o  e.  om
32elexi 2797 . . . . 5  |-  2o  e.  _V
4 domrefg 6896 . . . . 5  |-  ( 2o  e.  _V  ->  2o  ~<_  2o )
53cfpwsdom 8206 . . . . 5  |-  ( 2o  ~<_  2o  ->  ( aleph `  (/) )  ~<  ( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) ) )
63, 4, 5mp2b 9 . . . 4  |-  ( aleph `  (/) )  ~<  ( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )
7 aleph0 7693 . . . . . 6  |-  ( aleph `  (/) )  =  om
87a1i 10 . . . . 5  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( aleph `  (/) )  =  om )
97oveq2i 5869 . . . . . . . . . 10  |-  ( 2o 
^m  ( aleph `  (/) ) )  =  ( 2o  ^m  om )
109fveq2i 5528 . . . . . . . . 9  |-  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) )  =  ( card `  ( 2o  ^m  om ) )
1110eqeq1i 2290 . . . . . . . 8  |-  ( (
card `  ( 2o  ^m  ( aleph `  (/) ) ) )  =  ( aleph ` 
om )  <->  ( card `  ( 2o  ^m  om ) )  =  (
aleph `  om ) )
1211biimpri 197 . . . . . . 7  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( card `  ( 2o  ^m  ( aleph `  (/) ) ) )  =  ( aleph ` 
om ) )
1312fveq2d 5529 . . . . . 6  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )  =  ( cf `  ( aleph ` 
om ) ) )
14 limom 4671 . . . . . . . 8  |-  Lim  om
15 alephsing 7902 . . . . . . . 8  |-  ( Lim 
om  ->  ( cf `  ( aleph `  om ) )  =  ( cf `  om ) )
1614, 15ax-mp 8 . . . . . . 7  |-  ( cf `  ( aleph `  om ) )  =  ( cf `  om )
17 cfom 7890 . . . . . . 7  |-  ( cf ` 
om )  =  om
1816, 17eqtri 2303 . . . . . 6  |-  ( cf `  ( aleph `  om ) )  =  om
1913, 18syl6eq 2331 . . . . 5  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )  =  om )
208, 19breq12d 4036 . . . 4  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( ( aleph `  (/) )  ~< 
( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )  <->  om  ~<  om )
)
216, 20mpbii 202 . . 3  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  ->  om  ~<  om )
2221necon3bi 2487 . 2  |-  ( -. 
om  ~<  om  ->  ( card `  ( 2o  ^m  om ) )  =/=  ( aleph `  om ) )
231, 22ax-mp 8 1  |-  ( card `  ( 2o  ^m  om ) )  =/=  ( aleph `  om )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   class class class wbr 4023   Lim wlim 4393   omcom 4656   ` cfv 5255  (class class class)co 5858   2oc2o 6473    ^m cmap 6772    ~<_ cdom 6861    ~< csdm 6862   cardccrd 7568   alephcale 7569   cfccf 7570
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-ac2 8089
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-smo 6363  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-card 7572  df-aleph 7573  df-cf 7574  df-acn 7575  df-ac 7743
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