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Theorem alephom 8354
Description: From canth2 7157, 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 8338 (in the form of cfpwsdom 8353), 
( 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 7141 . 2  |-  -.  om  ~<  om
2 2onn 6780 . . . . . 6  |-  2o  e.  om
32elexi 2882 . . . . 5  |-  2o  e.  _V
4 domrefg 7039 . . . . 5  |-  ( 2o  e.  _V  ->  2o  ~<_  2o )
53cfpwsdom 8353 . . . . 5  |-  ( 2o  ~<_  2o  ->  ( aleph `  (/) )  ~<  ( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) ) )
63, 4, 5mp2b 9 . . . 4  |-  ( aleph `  (/) )  ~<  ( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )
7 aleph0 7840 . . . . . 6  |-  ( aleph `  (/) )  =  om
87a1i 10 . . . . 5  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( aleph `  (/) )  =  om )
97oveq2i 5992 . . . . . . . . . 10  |-  ( 2o 
^m  ( aleph `  (/) ) )  =  ( 2o  ^m  om )
109fveq2i 5635 . . . . . . . . 9  |-  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) )  =  ( card `  ( 2o  ^m  om ) )
1110eqeq1i 2373 . . . . . . . 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 5636 . . . . . 6  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )  =  ( cf `  ( aleph ` 
om ) ) )
14 limom 4774 . . . . . . . 8  |-  Lim  om
15 alephsing 8049 . . . . . . . 8  |-  ( Lim 
om  ->  ( cf `  ( aleph `  om ) )  =  ( cf `  om ) )
1614, 15ax-mp 8 . . . . . . 7  |-  ( cf `  ( aleph `  om ) )  =  ( cf `  om )
17 cfom 8037 . . . . . . 7  |-  ( cf ` 
om )  =  om
1816, 17eqtri 2386 . . . . . 6  |-  ( cf `  ( aleph `  om ) )  =  om
1913, 18syl6eq 2414 . . . . 5  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )  =  om )
208, 19breq12d 4138 . . . 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 2570 . 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 1647    e. wcel 1715    =/= wne 2529   _Vcvv 2873   (/)c0 3543   class class class wbr 4125   Lim wlim 4496   omcom 4759   ` cfv 5358  (class class class)co 5981   2oc2o 6615    ^m cmap 6915    ~<_ cdom 7004    ~< csdm 7005   cardccrd 7715   alephcale 7716   cfccf 7717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-ac2 8236
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-smo 6505  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-map 6917  df-ixp 6961  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-oi 7372  df-har 7419  df-card 7719  df-aleph 7720  df-cf 7721  df-acn 7722  df-ac 7890
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