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Theorem alephom 9392
Description: From canth2 8098, 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 9376 (in the form of cfpwsdom 9391), (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 8082 . 2 ¬ ω ≺ ω
2 2onn 7705 . . . . . 6 2𝑜 ∈ ω
32elexi 3208 . . . . 5 2𝑜 ∈ V
4 domrefg 7975 . . . . 5 (2𝑜 ∈ V → 2𝑜 ≼ 2𝑜)
53cfpwsdom 9391 . . . . 5 (2𝑜 ≼ 2𝑜 → (ℵ‘∅) ≺ (cf‘(card‘(2𝑜𝑚 (ℵ‘∅)))))
63, 4, 5mp2b 10 . . . 4 (ℵ‘∅) ≺ (cf‘(card‘(2𝑜𝑚 (ℵ‘∅))))
7 aleph0 8874 . . . . . 6 (ℵ‘∅) = ω
87a1i 11 . . . . 5 ((card‘(2𝑜𝑚 ω)) = (ℵ‘ω) → (ℵ‘∅) = ω)
97oveq2i 6646 . . . . . . . . . 10 (2𝑜𝑚 (ℵ‘∅)) = (2𝑜𝑚 ω)
109fveq2i 6181 . . . . . . . . 9 (card‘(2𝑜𝑚 (ℵ‘∅))) = (card‘(2𝑜𝑚 ω))
1110eqeq1i 2625 . . . . . . . 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 6182 . . . . . 6 ((card‘(2𝑜𝑚 ω)) = (ℵ‘ω) → (cf‘(card‘(2𝑜𝑚 (ℵ‘∅)))) = (cf‘(ℵ‘ω)))
14 limom 7065 . . . . . . . 8 Lim ω
15 alephsing 9083 . . . . . . . 8 (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω))
1614, 15ax-mp 5 . . . . . . 7 (cf‘(ℵ‘ω)) = (cf‘ω)
17 cfom 9071 . . . . . . 7 (cf‘ω) = ω
1816, 17eqtri 2642 . . . . . 6 (cf‘(ℵ‘ω)) = ω
1913, 18syl6eq 2670 . . . . 5 ((card‘(2𝑜𝑚 ω)) = (ℵ‘ω) → (cf‘(card‘(2𝑜𝑚 (ℵ‘∅)))) = ω)
208, 19breq12d 4657 . . . 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 2817 . 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 1481  wcel 1988  wne 2791  Vcvv 3195  c0 3907   class class class wbr 4644  Lim wlim 5712  cfv 5876  (class class class)co 6635  ωcom 7050  2𝑜c2o 7539  𝑚 cmap 7842  cdom 7938  csdm 7939  cardccrd 8746  cale 8747  cfccf 8748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-ac2 9270
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-smo 7428  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-ixp 7894  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-oi 8400  df-har 8448  df-card 8750  df-aleph 8751  df-cf 8752  df-acn 8753  df-ac 8924
This theorem is referenced by: (None)
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