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Theorem alephom 8444
Description: From canth2 7246, 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 8428 (in the form of cfpwsdom 8443), 
( 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 7230 . 2  |-  -.  om  ~<  om
2 2onn 6869 . . . . . 6  |-  2o  e.  om
32elexi 2952 . . . . 5  |-  2o  e.  _V
4 domrefg 7128 . . . . 5  |-  ( 2o  e.  _V  ->  2o  ~<_  2o )
53cfpwsdom 8443 . . . . 5  |-  ( 2o  ~<_  2o  ->  ( aleph `  (/) )  ~<  ( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) ) )
63, 4, 5mp2b 10 . . . 4  |-  ( aleph `  (/) )  ~<  ( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )
7 aleph0 7931 . . . . . 6  |-  ( aleph `  (/) )  =  om
87a1i 11 . . . . 5  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( aleph `  (/) )  =  om )
97oveq2i 6078 . . . . . . . . . 10  |-  ( 2o 
^m  ( aleph `  (/) ) )  =  ( 2o  ^m  om )
109fveq2i 5717 . . . . . . . . 9  |-  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) )  =  ( card `  ( 2o  ^m  om ) )
1110eqeq1i 2437 . . . . . . . 8  |-  ( (
card `  ( 2o  ^m  ( aleph `  (/) ) ) )  =  ( aleph ` 
om )  <->  ( card `  ( 2o  ^m  om ) )  =  (
aleph `  om ) )
1211biimpri 198 . . . . . . 7  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( card `  ( 2o  ^m  ( aleph `  (/) ) ) )  =  ( aleph ` 
om ) )
1312fveq2d 5718 . . . . . 6  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )  =  ( cf `  ( aleph ` 
om ) ) )
14 limom 4846 . . . . . . . 8  |-  Lim  om
15 alephsing 8140 . . . . . . . 8  |-  ( Lim 
om  ->  ( cf `  ( aleph `  om ) )  =  ( cf `  om ) )
1614, 15ax-mp 8 . . . . . . 7  |-  ( cf `  ( aleph `  om ) )  =  ( cf `  om )
17 cfom 8128 . . . . . . 7  |-  ( cf ` 
om )  =  om
1816, 17eqtri 2450 . . . . . 6  |-  ( cf `  ( aleph `  om ) )  =  om
1913, 18syl6eq 2478 . . . . 5  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )  =  om )
208, 19breq12d 4212 . . . 4  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( ( aleph `  (/) )  ~< 
( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )  <->  om  ~<  om )
)
216, 20mpbii 203 . . 3  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  ->  om  ~<  om )
2221necon3bi 2634 . 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 1652    e. wcel 1725    =/= wne 2593   _Vcvv 2943   (/)c0 3615   class class class wbr 4199   Lim wlim 4569   omcom 4831   ` cfv 5440  (class class class)co 6067   2oc2o 6704    ^m cmap 7004    ~<_ cdom 7093    ~< csdm 7094   cardccrd 7806   alephcale 7807   cfccf 7808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-inf2 7580  ax-ac2 8327
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-iin 4083  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-se 4529  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-isom 5449  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-riota 6535  df-smo 6594  df-recs 6619  df-rdg 6654  df-1o 6710  df-2o 6711  df-oadd 6714  df-er 6891  df-map 7006  df-ixp 7050  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-oi 7463  df-har 7510  df-card 7810  df-aleph 7811  df-cf 7812  df-acn 7813  df-ac 7981
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