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Theorem alephom 8223
 Description: From canth2 7030, we know that , but we cannot prove that (this is the Continuum Hypothesis), nor can we prove that it is less than any bound whatsoever (i.e. the statement is consistent for any ordinal ). However, we can prove that is not equal to , nor , on cofinality grounds, because by Konig's Theorem konigth 8207 (in the form of cfpwsdom 8222), has uncountable cofinality, which eliminates limit alephs like . (The first limit aleph that is not eliminated is , which has cofinality .) (Contributed by Mario Carneiro, 21-Mar-2013.)
Assertion
Ref Expression
alephom

Proof of Theorem alephom
StepHypRef Expression
1 sdomirr 7014 . 2
2 2onn 6654 . . . . . 6
32elexi 2810 . . . . 5
4 domrefg 6912 . . . . 5
53cfpwsdom 8222 . . . . 5
63, 4, 5mp2b 9 . . . 4
7 aleph0 7709 . . . . . 6
87a1i 10 . . . . 5
97oveq2i 5885 . . . . . . . . . 10
109fveq2i 5544 . . . . . . . . 9
1110eqeq1i 2303 . . . . . . . 8
1211biimpri 197 . . . . . . 7
1312fveq2d 5545 . . . . . 6
14 limom 4687 . . . . . . . 8
15 alephsing 7918 . . . . . . . 8
1614, 15ax-mp 8 . . . . . . 7
17 cfom 7906 . . . . . . 7
1816, 17eqtri 2316 . . . . . 6
1913, 18syl6eq 2344 . . . . 5
208, 19breq12d 4052 . . . 4
216, 20mpbii 202 . . 3
2221necon3bi 2500 . 2
231, 22ax-mp 8 1
 Colors of variables: wff set class Syntax hints:   wn 3   wceq 1632   wcel 1696   wne 2459  cvv 2801  c0 3468   class class class wbr 4039   wlim 4409  com 4672  cfv 5271  (class class class)co 5874  c2o 6489   cmap 6788   cdom 6877   csdm 6878  ccrd 7584  cale 7585  ccf 7586 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-ac2 8105 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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-smo 6379  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-card 7588  df-aleph 7589  df-cf 7590  df-acn 7591  df-ac 7759
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