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Theorem alephom 9608
Description: From canth2 8268, 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 9592 (in the form of cfpwsdom 9607), (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 8252 . 2 ¬ ω ≺ ω
2 2onn 7873 . . . . . 6 2𝑜 ∈ ω
32elexi 3362 . . . . 5 2𝑜 ∈ V
4 domrefg 8143 . . . . 5 (2𝑜 ∈ V → 2𝑜 ≼ 2𝑜)
53cfpwsdom 9607 . . . . 5 (2𝑜 ≼ 2𝑜 → (ℵ‘∅) ≺ (cf‘(card‘(2𝑜𝑚 (ℵ‘∅)))))
63, 4, 5mp2b 10 . . . 4 (ℵ‘∅) ≺ (cf‘(card‘(2𝑜𝑚 (ℵ‘∅))))
7 aleph0 9088 . . . . . 6 (ℵ‘∅) = ω
87a1i 11 . . . . 5 ((card‘(2𝑜𝑚 ω)) = (ℵ‘ω) → (ℵ‘∅) = ω)
97oveq2i 6803 . . . . . . . . . 10 (2𝑜𝑚 (ℵ‘∅)) = (2𝑜𝑚 ω)
109fveq2i 6335 . . . . . . . . 9 (card‘(2𝑜𝑚 (ℵ‘∅))) = (card‘(2𝑜𝑚 ω))
1110eqeq1i 2775 . . . . . . . 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 6336 . . . . . 6 ((card‘(2𝑜𝑚 ω)) = (ℵ‘ω) → (cf‘(card‘(2𝑜𝑚 (ℵ‘∅)))) = (cf‘(ℵ‘ω)))
14 limom 7226 . . . . . . . 8 Lim ω
15 alephsing 9299 . . . . . . . 8 (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω))
1614, 15ax-mp 5 . . . . . . 7 (cf‘(ℵ‘ω)) = (cf‘ω)
17 cfom 9287 . . . . . . 7 (cf‘ω) = ω
1816, 17eqtri 2792 . . . . . 6 (cf‘(ℵ‘ω)) = ω
1913, 18syl6eq 2820 . . . . 5 ((card‘(2𝑜𝑚 ω)) = (ℵ‘ω) → (cf‘(card‘(2𝑜𝑚 (ℵ‘∅)))) = ω)
208, 19breq12d 4797 . . . 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 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 1630  wcel 2144  wne 2942  Vcvv 3349  c0 4061   class class class wbr 4784  Lim wlim 5867  cfv 6031  (class class class)co 6792  ωcom 7211  2𝑜c2o 7706  𝑚 cmap 8008  cdom 8106  csdm 8107  cardccrd 8960  cale 8961  cfccf 8962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-ac2 9486
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-smo 7595  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-er 7895  df-map 8010  df-ixp 8062  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-oi 8570  df-har 8618  df-card 8964  df-aleph 8965  df-cf 8966  df-acn 8967  df-ac 9138
This theorem is referenced by: (None)
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