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Theorem alephom 10626
Description: From canth2 9171, 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 10610 (in the form of cfpwsdom 10625), (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 9155 . 2 ¬ ω ≺ ω
2 2onn 8681 . . . . . 6 2o ∈ ω
32elexi 3502 . . . . 5 2o ∈ V
4 domrefg 9028 . . . . 5 (2o ∈ V → 2o ≼ 2o)
53cfpwsdom 10625 . . . . 5 (2o ≼ 2o → (ℵ‘∅) ≺ (cf‘(card‘(2om (ℵ‘∅)))))
63, 4, 5mp2b 10 . . . 4 (ℵ‘∅) ≺ (cf‘(card‘(2om (ℵ‘∅))))
7 aleph0 10107 . . . . . 6 (ℵ‘∅) = ω
87a1i 11 . . . . 5 ((card‘(2om ω)) = (ℵ‘ω) → (ℵ‘∅) = ω)
97oveq2i 7443 . . . . . . . . . 10 (2om (ℵ‘∅)) = (2om ω)
109fveq2i 6908 . . . . . . . . 9 (card‘(2om (ℵ‘∅))) = (card‘(2om ω))
1110eqeq1i 2741 . . . . . . . 8 ((card‘(2om (ℵ‘∅))) = (ℵ‘ω) ↔ (card‘(2om ω)) = (ℵ‘ω))
1211biimpri 228 . . . . . . 7 ((card‘(2om ω)) = (ℵ‘ω) → (card‘(2om (ℵ‘∅))) = (ℵ‘ω))
1312fveq2d 6909 . . . . . 6 ((card‘(2om ω)) = (ℵ‘ω) → (cf‘(card‘(2om (ℵ‘∅)))) = (cf‘(ℵ‘ω)))
14 limom 7904 . . . . . . . 8 Lim ω
15 alephsing 10317 . . . . . . . 8 (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω))
1614, 15ax-mp 5 . . . . . . 7 (cf‘(ℵ‘ω)) = (cf‘ω)
17 cfom 10305 . . . . . . 7 (cf‘ω) = ω
1816, 17eqtri 2764 . . . . . 6 (cf‘(ℵ‘ω)) = ω
1913, 18eqtrdi 2792 . . . . 5 ((card‘(2om ω)) = (ℵ‘ω) → (cf‘(card‘(2om (ℵ‘∅)))) = ω)
208, 19breq12d 5155 . . . 4 ((card‘(2om ω)) = (ℵ‘ω) → ((ℵ‘∅) ≺ (cf‘(card‘(2om (ℵ‘∅)))) ↔ ω ≺ ω))
216, 20mpbii 233 . . 3 ((card‘(2om ω)) = (ℵ‘ω) → ω ≺ ω)
2221necon3bi 2966 . 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 1539  wcel 2107  wne 2939  Vcvv 3479  c0 4332   class class class wbr 5142  Lim wlim 6384  cfv 6560  (class class class)co 7432  ωcom 7888  2oc2o 8501  m cmap 8867  cdom 8984  csdm 8985  cardccrd 9976  cale 9977  cfccf 9978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-ac2 10504
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-smo 8387  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-oi 9551  df-har 9598  df-card 9980  df-aleph 9981  df-cf 9982  df-acn 9983  df-ac 10157
This theorem is referenced by: (None)
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