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Theorem aleph1 4843
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.)
Assertion
Ref Expression
aleph1 |- (aleph` 1o) ~<_ (2o ^m (aleph` (/)))

Proof of Theorem aleph1
StepHypRef Expression
1 df-1o 4117 . . 3 |- 1o = suc (/)
21fveq2i 3712 . 2 |- (aleph` 1o) = (aleph` suc (/))
3 alephsucpw 4842 . . 3 |- (aleph` suc (/)) ~<_ P~(aleph` (/))
4 oprex 3968 . . . 4 |- (2o ^m (aleph` (/))) e. V
5 fvex 3717 . . . . 5 |- (aleph` (/)) e. V
65pw2en 4426 . . . 4 |- P~(aleph` (/)) ~~ (2o ^m (aleph` (/)))
7 domen2 4460 . . . 4 |- (((2o ^m (aleph` (/))) e. V /\ P~(aleph` (/)) ~~ (2o ^m (aleph` (/)))) -> ((aleph` suc (/)) ~<_ P~(aleph` (/)) <-> (aleph` suc (/)) ~<_ (2o ^m (aleph` (/)))))
84, 6, 7mp2an 695 . . 3 |- ((aleph` suc (/)) ~<_ P~(aleph` (/)) <-> (aleph` suc (/)) ~<_ (2o ^m (aleph` (/))))
93, 8mpbi 189 . 2 |- (aleph` suc (/)) ~<_ (2o ^m (aleph` (/)))
102, 9eqbrtr 2624 1 |- (aleph` 1o) ~<_ (2o ^m (aleph` (/)))
Colors of variables: wff set class
Syntax hints:   <-> wb 146   e. wcel 955  Vcvv 1802  (/)c0 2270  P~cpw 2391   class class class wbr 2609  suc csuc 2940  ` cfv 3172  (class class class)co 3948  1oc1o 4112  2oc2o 4113   ^m cm 4306   ~~ cen 4348   ~<_ cdom 4349  alephcale 4786
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597  ax-ac 4716
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1o 4117  df-2o 4118  df-er 4245  df-map 4308  df-en 4351  df-dom 4352  df-sdom 4353  df-card 4788  df-aleph 4789
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