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Theorem alephom 10623
Description: From canth2 9169, 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 10607 (in the form of cfpwsdom 10622), (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 9153 . 2 ¬ ω ≺ ω
2 2onn 8679 . . . . . 6 2o ∈ ω
32elexi 3501 . . . . 5 2o ∈ V
4 domrefg 9026 . . . . 5 (2o ∈ V → 2o ≼ 2o)
53cfpwsdom 10622 . . . . 5 (2o ≼ 2o → (ℵ‘∅) ≺ (cf‘(card‘(2om (ℵ‘∅)))))
63, 4, 5mp2b 10 . . . 4 (ℵ‘∅) ≺ (cf‘(card‘(2om (ℵ‘∅))))
7 aleph0 10104 . . . . . 6 (ℵ‘∅) = ω
87a1i 11 . . . . 5 ((card‘(2om ω)) = (ℵ‘ω) → (ℵ‘∅) = ω)
97oveq2i 7442 . . . . . . . . . 10 (2om (ℵ‘∅)) = (2om ω)
109fveq2i 6910 . . . . . . . . 9 (card‘(2om (ℵ‘∅))) = (card‘(2om ω))
1110eqeq1i 2740 . . . . . . . 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 6911 . . . . . 6 ((card‘(2om ω)) = (ℵ‘ω) → (cf‘(card‘(2om (ℵ‘∅)))) = (cf‘(ℵ‘ω)))
14 limom 7903 . . . . . . . 8 Lim ω
15 alephsing 10314 . . . . . . . 8 (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω))
1614, 15ax-mp 5 . . . . . . 7 (cf‘(ℵ‘ω)) = (cf‘ω)
17 cfom 10302 . . . . . . 7 (cf‘ω) = ω
1816, 17eqtri 2763 . . . . . 6 (cf‘(ℵ‘ω)) = ω
1913, 18eqtrdi 2791 . . . . 5 ((card‘(2om ω)) = (ℵ‘ω) → (cf‘(card‘(2om (ℵ‘∅)))) = ω)
208, 19breq12d 5161 . . . 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 2965 . 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 1537  wcel 2106  wne 2938  Vcvv 3478  c0 4339   class class class wbr 5148  Lim wlim 6387  cfv 6563  (class class class)co 7431  ωcom 7887  2oc2o 8499  m cmap 8865  cdom 8982  csdm 8983  cardccrd 9973  cale 9974  cfccf 9975
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-smo 8385  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-har 9595  df-card 9977  df-aleph 9978  df-cf 9979  df-acn 9980  df-ac 10154
This theorem is referenced by: (None)
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