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Theorem alephom 10529
Description: From canth2 9080, 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 10513 (in the form of cfpwsdom 10528), (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 9064 . 2 ¬ ω ≺ ω
2 2onn 8592 . . . . . 6 2o ∈ ω
32elexi 3466 . . . . 5 2o ∈ V
4 domrefg 8933 . . . . 5 (2o ∈ V → 2o ≼ 2o)
53cfpwsdom 10528 . . . . 5 (2o ≼ 2o → (ℵ‘∅) ≺ (cf‘(card‘(2om (ℵ‘∅)))))
63, 4, 5mp2b 10 . . . 4 (ℵ‘∅) ≺ (cf‘(card‘(2om (ℵ‘∅))))
7 aleph0 10010 . . . . . 6 (ℵ‘∅) = ω
87a1i 11 . . . . 5 ((card‘(2om ω)) = (ℵ‘ω) → (ℵ‘∅) = ω)
97oveq2i 7372 . . . . . . . . . 10 (2om (ℵ‘∅)) = (2om ω)
109fveq2i 6849 . . . . . . . . 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 6850 . . . . . 6 ((card‘(2om ω)) = (ℵ‘ω) → (cf‘(card‘(2om (ℵ‘∅)))) = (cf‘(ℵ‘ω)))
14 limom 7822 . . . . . . . 8 Lim ω
15 alephsing 10220 . . . . . . . 8 (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω))
1614, 15ax-mp 5 . . . . . . 7 (cf‘(ℵ‘ω)) = (cf‘ω)
17 cfom 10208 . . . . . . 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 5122 . . . 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 2967 . 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 2940  Vcvv 3447  c0 4286   class class class wbr 5109  Lim wlim 6322  cfv 6500  (class class class)co 7361  ωcom 7806  2oc2o 8410  m cmap 8771  cdom 8887  csdm 8888  cardccrd 9879  cale 9880  cfccf 9881
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-inf2 9585  ax-ac2 10407
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 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-smo 8296  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-er 8654  df-map 8773  df-ixp 8842  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-oi 9454  df-har 9501  df-card 9883  df-aleph 9884  df-cf 9885  df-acn 9886  df-ac 10060
This theorem is referenced by: (None)
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