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Theorem aleph1 8193
Description: The set exponentiation of 2 to the aleph-zero has cardinality of at least aleph-one. (If we were to assume the Continuum Hypothesis, their cardinalities would be the same.) (Contributed by NM, 7-Jul-2004.)
Assertion
Ref Expression
aleph1  |-  ( aleph `  1o )  ~<_  ( 2o 
^m  ( aleph `  (/) ) )

Proof of Theorem aleph1
StepHypRef Expression
1 df-1o 6479 . . 3  |-  1o  =  suc  (/)
21fveq2i 5528 . 2  |-  ( aleph `  1o )  =  (
aleph `  suc  (/) )
3 alephsucpw 8192 . . 3  |-  ( aleph ` 
suc  (/) )  ~<_  ~P ( aleph `  (/) )
4 fvex 5539 . . . . 5  |-  ( aleph `  (/) )  e.  _V
54pw2en 6969 . . . 4  |-  ~P ( aleph `  (/) )  ~~  ( 2o  ^m  ( aleph `  (/) ) )
6 domen2 7004 . . . 4  |-  ( ~P ( aleph `  (/) )  ~~  ( 2o  ^m  ( aleph `  (/) ) )  -> 
( ( aleph `  suc  (/) )  ~<_  ~P ( aleph `  (/) )  <->  ( aleph ` 
suc  (/) )  ~<_  ( 2o 
^m  ( aleph `  (/) ) ) ) )
75, 6ax-mp 8 . . 3  |-  ( (
aleph `  suc  (/) )  ~<_  ~P ( aleph `  (/) )  <->  ( aleph ` 
suc  (/) )  ~<_  ( 2o 
^m  ( aleph `  (/) ) ) )
83, 7mpbi 199 . 2  |-  ( aleph ` 
suc  (/) )  ~<_  ( 2o 
^m  ( aleph `  (/) ) )
92, 8eqbrtri 4042 1  |-  ( aleph `  1o )  ~<_  ( 2o 
^m  ( aleph `  (/) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   (/)c0 3455   ~Pcpw 3625   class class class wbr 4023   suc csuc 4394   ` cfv 5255  (class class class)co 5858   1oc1o 6472   2oc2o 6473    ^m cmap 6772    ~~ cen 6860    ~<_ cdom 6861   alephcale 7569
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-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-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-er 6660  df-map 6774  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-ac 7743
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