MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alephom Structured version   Visualization version   GIF version

Theorem alephom 9354
Description: From canth2 8060, 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 9338 (in the form of cfpwsdom 9353), (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 8044 . 2 ¬ ω ≺ ω
2 2onn 7668 . . . . . 6 2𝑜 ∈ ω
32elexi 3199 . . . . 5 2𝑜 ∈ V
4 domrefg 7937 . . . . 5 (2𝑜 ∈ V → 2𝑜 ≼ 2𝑜)
53cfpwsdom 9353 . . . . 5 (2𝑜 ≼ 2𝑜 → (ℵ‘∅) ≺ (cf‘(card‘(2𝑜𝑚 (ℵ‘∅)))))
63, 4, 5mp2b 10 . . . 4 (ℵ‘∅) ≺ (cf‘(card‘(2𝑜𝑚 (ℵ‘∅))))
7 aleph0 8836 . . . . . 6 (ℵ‘∅) = ω
87a1i 11 . . . . 5 ((card‘(2𝑜𝑚 ω)) = (ℵ‘ω) → (ℵ‘∅) = ω)
97oveq2i 6618 . . . . . . . . . 10 (2𝑜𝑚 (ℵ‘∅)) = (2𝑜𝑚 ω)
109fveq2i 6153 . . . . . . . . 9 (card‘(2𝑜𝑚 (ℵ‘∅))) = (card‘(2𝑜𝑚 ω))
1110eqeq1i 2626 . . . . . . . 8 ((card‘(2𝑜𝑚 (ℵ‘∅))) = (ℵ‘ω) ↔ (card‘(2𝑜𝑚 ω)) = (ℵ‘ω))
1211biimpri 218 . . . . . . 7 ((card‘(2𝑜𝑚 ω)) = (ℵ‘ω) → (card‘(2𝑜𝑚 (ℵ‘∅))) = (ℵ‘ω))
1312fveq2d 6154 . . . . . 6 ((card‘(2𝑜𝑚 ω)) = (ℵ‘ω) → (cf‘(card‘(2𝑜𝑚 (ℵ‘∅)))) = (cf‘(ℵ‘ω)))
14 limom 7030 . . . . . . . 8 Lim ω
15 alephsing 9045 . . . . . . . 8 (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω))
1614, 15ax-mp 5 . . . . . . 7 (cf‘(ℵ‘ω)) = (cf‘ω)
17 cfom 9033 . . . . . . 7 (cf‘ω) = ω
1816, 17eqtri 2643 . . . . . 6 (cf‘(ℵ‘ω)) = ω
1913, 18syl6eq 2671 . . . . 5 ((card‘(2𝑜𝑚 ω)) = (ℵ‘ω) → (cf‘(card‘(2𝑜𝑚 (ℵ‘∅)))) = ω)
208, 19breq12d 4628 . . . 4 ((card‘(2𝑜𝑚 ω)) = (ℵ‘ω) → ((ℵ‘∅) ≺ (cf‘(card‘(2𝑜𝑚 (ℵ‘∅)))) ↔ ω ≺ ω))
216, 20mpbii 223 . . 3 ((card‘(2𝑜𝑚 ω)) = (ℵ‘ω) → ω ≺ ω)
2221necon3bi 2816 . 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 1480  wcel 1987  wne 2790  Vcvv 3186  c0 3893   class class class wbr 4615  Lim wlim 5685  cfv 5849  (class class class)co 6607  ωcom 7015  2𝑜c2o 7502  𝑚 cmap 7805  cdom 7900  csdm 7901  cardccrd 8708  cale 8709  cfccf 8710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-inf2 8485  ax-ac2 9232
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-iin 4490  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-smo 7391  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-er 7690  df-map 7807  df-ixp 7856  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-oi 8362  df-har 8410  df-card 8712  df-aleph 8713  df-cf 8714  df-acn 8715  df-ac 8886
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator