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Theorem aleph1 8937
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 7122 . . 3  |-  1o  =  suc  (/)
21fveq2i 5851 . 2  |-  ( aleph `  1o )  =  (
aleph `  suc  (/) )
3 alephsucpw 8936 . . 3  |-  ( aleph ` 
suc  (/) )  ~<_  ~P ( aleph `  (/) )
4 fvex 5858 . . . . 5  |-  ( aleph `  (/) )  e.  _V
54pw2en 7617 . . . 4  |-  ~P ( aleph `  (/) )  ~~  ( 2o  ^m  ( aleph `  (/) ) )
6 domen2 7653 . . . 4  |-  ( ~P ( aleph `  (/) )  ~~  ( 2o  ^m  ( aleph `  (/) ) )  -> 
( ( aleph `  suc  (/) )  ~<_  ~P ( aleph `  (/) )  <->  ( aleph ` 
suc  (/) )  ~<_  ( 2o 
^m  ( aleph `  (/) ) ) ) )
75, 6ax-mp 5 . . 3  |-  ( (
aleph `  suc  (/) )  ~<_  ~P ( aleph `  (/) )  <->  ( aleph ` 
suc  (/) )  ~<_  ( 2o 
^m  ( aleph `  (/) ) ) )
83, 7mpbi 208 . 2  |-  ( aleph ` 
suc  (/) )  ~<_  ( 2o 
^m  ( aleph `  (/) ) )
92, 8eqbrtri 4458 1  |-  ( aleph `  1o )  ~<_  ( 2o 
^m  ( aleph `  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   (/)c0 3783   ~Pcpw 3999   class class class wbr 4439   suc csuc 4869   ` cfv 5570  (class class class)co 6270   1oc1o 7115   2oc2o 7116    ^m cmap 7412    ~~ cen 7506    ~<_ cdom 7507   alephcale 8308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-ac2 8834
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-oi 7927  df-har 7976  df-card 8311  df-aleph 8312  df-ac 8488
This theorem is referenced by: (None)
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