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Theorem alephom 10580
Description: From canth2 9130, 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 10564 (in the form of cfpwsdom 10579), (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 9114 . 2 ¬ ω ≺ ω
2 2onn 8641 . . . . . 6 2o ∈ ω
32elexi 3494 . . . . 5 2o ∈ V
4 domrefg 8983 . . . . 5 (2o ∈ V → 2o ≼ 2o)
53cfpwsdom 10579 . . . . 5 (2o ≼ 2o → (ℵ‘∅) ≺ (cf‘(card‘(2om (ℵ‘∅)))))
63, 4, 5mp2b 10 . . . 4 (ℵ‘∅) ≺ (cf‘(card‘(2om (ℵ‘∅))))
7 aleph0 10061 . . . . . 6 (ℵ‘∅) = ω
87a1i 11 . . . . 5 ((card‘(2om ω)) = (ℵ‘ω) → (ℵ‘∅) = ω)
97oveq2i 7420 . . . . . . . . . 10 (2om (ℵ‘∅)) = (2om ω)
109fveq2i 6895 . . . . . . . . 9 (card‘(2om (ℵ‘∅))) = (card‘(2om ω))
1110eqeq1i 2738 . . . . . . . 8 ((card‘(2om (ℵ‘∅))) = (ℵ‘ω) ↔ (card‘(2om ω)) = (ℵ‘ω))
1211biimpri 227 . . . . . . 7 ((card‘(2om ω)) = (ℵ‘ω) → (card‘(2om (ℵ‘∅))) = (ℵ‘ω))
1312fveq2d 6896 . . . . . 6 ((card‘(2om ω)) = (ℵ‘ω) → (cf‘(card‘(2om (ℵ‘∅)))) = (cf‘(ℵ‘ω)))
14 limom 7871 . . . . . . . 8 Lim ω
15 alephsing 10271 . . . . . . . 8 (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω))
1614, 15ax-mp 5 . . . . . . 7 (cf‘(ℵ‘ω)) = (cf‘ω)
17 cfom 10259 . . . . . . 7 (cf‘ω) = ω
1816, 17eqtri 2761 . . . . . 6 (cf‘(ℵ‘ω)) = ω
1913, 18eqtrdi 2789 . . . . 5 ((card‘(2om ω)) = (ℵ‘ω) → (cf‘(card‘(2om (ℵ‘∅)))) = ω)
208, 19breq12d 5162 . . . 4 ((card‘(2om ω)) = (ℵ‘ω) → ((ℵ‘∅) ≺ (cf‘(card‘(2om (ℵ‘∅)))) ↔ ω ≺ ω))
216, 20mpbii 232 . . 3 ((card‘(2om ω)) = (ℵ‘ω) → ω ≺ ω)
2221necon3bi 2968 . 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 1542  wcel 2107  wne 2941  Vcvv 3475  c0 4323   class class class wbr 5149  Lim wlim 6366  cfv 6544  (class class class)co 7409  ωcom 7855  2oc2o 8460  m cmap 8820  cdom 8937  csdm 8938  cardccrd 9930  cale 9931  cfccf 9932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-ac2 10458
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-smo 8346  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9505  df-har 9552  df-card 9934  df-aleph 9935  df-cf 9936  df-acn 9937  df-ac 10111
This theorem is referenced by: (None)
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