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Theorem alephom 10001
 Description: From canth2 8664, we know that (ℵ‘0) < (2↑ω), but we cannot prove that (2↑ω) = (ℵ‘1) (this is the Continuum Hypothesis), nor can we prove that it is less than any bound whatsoever (i.e. the statement (ℵ‘𝐴) < (2↑ω) is consistent for any ordinal 𝐴). However, we can prove that (2↑ω) is not equal to (ℵ‘ω), nor (ℵ‘(ℵ‘ω)), on cofinality grounds, because by Konig's Theorem konigth 9985 (in the form of cfpwsdom 10000), (2↑ω) has uncountable cofinality, which eliminates limit alephs like (ℵ‘ω). (The first limit aleph that is not eliminated is (ℵ‘(ℵ‘1)), which has cofinality (ℵ‘1).) (Contributed by Mario Carneiro, 21-Mar-2013.)
Assertion
Ref Expression
alephom (card‘(2om ω)) ≠ (ℵ‘ω)

Proof of Theorem alephom
StepHypRef Expression
1 sdomirr 8648 . 2 ¬ ω ≺ ω
2 2onn 8260 . . . . . 6 2o ∈ ω
32elexi 3513 . . . . 5 2o ∈ V
4 domrefg 8538 . . . . 5 (2o ∈ V → 2o ≼ 2o)
53cfpwsdom 10000 . . . . 5 (2o ≼ 2o → (ℵ‘∅) ≺ (cf‘(card‘(2om (ℵ‘∅)))))
63, 4, 5mp2b 10 . . . 4 (ℵ‘∅) ≺ (cf‘(card‘(2om (ℵ‘∅))))
7 aleph0 9486 . . . . . 6 (ℵ‘∅) = ω
87a1i 11 . . . . 5 ((card‘(2om ω)) = (ℵ‘ω) → (ℵ‘∅) = ω)
97oveq2i 7161 . . . . . . . . . 10 (2om (ℵ‘∅)) = (2om ω)
109fveq2i 6667 . . . . . . . . 9 (card‘(2om (ℵ‘∅))) = (card‘(2om ω))
1110eqeq1i 2826 . . . . . . . 8 ((card‘(2om (ℵ‘∅))) = (ℵ‘ω) ↔ (card‘(2om ω)) = (ℵ‘ω))
1211biimpri 230 . . . . . . 7 ((card‘(2om ω)) = (ℵ‘ω) → (card‘(2om (ℵ‘∅))) = (ℵ‘ω))
1312fveq2d 6668 . . . . . 6 ((card‘(2om ω)) = (ℵ‘ω) → (cf‘(card‘(2om (ℵ‘∅)))) = (cf‘(ℵ‘ω)))
14 limom 7589 . . . . . . . 8 Lim ω
15 alephsing 9692 . . . . . . . 8 (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω))
1614, 15ax-mp 5 . . . . . . 7 (cf‘(ℵ‘ω)) = (cf‘ω)
17 cfom 9680 . . . . . . 7 (cf‘ω) = ω
1816, 17eqtri 2844 . . . . . 6 (cf‘(ℵ‘ω)) = ω
1913, 18syl6eq 2872 . . . . 5 ((card‘(2om ω)) = (ℵ‘ω) → (cf‘(card‘(2om (ℵ‘∅)))) = ω)
208, 19breq12d 5071 . . . 4 ((card‘(2om ω)) = (ℵ‘ω) → ((ℵ‘∅) ≺ (cf‘(card‘(2om (ℵ‘∅)))) ↔ ω ≺ ω))
216, 20mpbii 235 . . 3 ((card‘(2om ω)) = (ℵ‘ω) → ω ≺ ω)
2221necon3bi 3042 . 2 (¬ ω ≺ ω → (card‘(2om ω)) ≠ (ℵ‘ω))
231, 22ax-mp 5 1 (card‘(2om ω)) ≠ (ℵ‘ω)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  Vcvv 3494  ∅c0 4290   class class class wbr 5058  Lim wlim 6186  ‘cfv 6349  (class class class)co 7150  ωcom 7574  2oc2o 8090   ↑m cmap 8400   ≼ cdom 8501   ≺ csdm 8502  cardccrd 9358  ℵcale 9359  cfccf 9360 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-inf2 9098  ax-ac2 9879 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-iin 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-smo 7977  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-oi 8968  df-har 9016  df-card 9362  df-aleph 9363  df-cf 9364  df-acn 9365  df-ac 9536 This theorem is referenced by: (None)
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