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Theorem unialeph 7744
Description: The union of the class of transfinite cardinals (the range of the aleph function) is the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
unialeph  |-  U. ran  aleph  =  On

Proof of Theorem unialeph
StepHypRef Expression
1 alephprc 7742 . . . 4  |-  -.  ran  aleph  e.  _V
2 uniexb 4579 . . . 4  |-  ( ran  aleph  e.  _V  <->  U. ran  aleph  e.  _V )
31, 2mtbi 289 . . 3  |-  -.  U. ran  aleph  e.  _V
4 elex 2809 . . 3  |-  ( U. ran  aleph  e.  On  ->  U.
ran  aleph  e.  _V )
53, 4mto 167 . 2  |-  -.  U. ran  aleph  e.  On
6 alephsson 7743 . . . . 5  |-  ran  aleph  C_  On
7 ssorduni 4593 . . . . 5  |-  ( ran  aleph 
C_  On  ->  Ord  U. ran  aleph )
86, 7ax-mp 8 . . . 4  |-  Ord  U. ran  aleph
9 ordeleqon 4596 . . . 4  |-  ( Ord  U. ran  aleph 
<->  ( U. ran  aleph  e.  On  \/  U. ran  aleph  =  On ) )
108, 9mpbi 199 . . 3  |-  ( U. ran  aleph  e.  On  \/  U.
ran  aleph  =  On )
1110ori 364 . 2  |-  ( -. 
U. ran  aleph  e.  On  ->  U. ran  aleph  =  On )
125, 11ax-mp 8 1  |-  U. ran  aleph  =  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   U.cuni 3843   Ord word 4407   Oncon0 4408   ran crn 4706   alephcale 7585
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
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-nel 2462  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-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-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-card 7588  df-aleph 7589
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