Theorem alephom 10009

Description: From canth2 8672, 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 9993 (in the form of cfpwsdom 10008), (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‘(2o ↑m ω)) ≠ (ℵ‘ω)

Proof of Theorem alephom

Step	Hyp	Ref	Expression
1		sdomirr 8656	. 2 ⊢ ¬ ω ≺ ω
2		2onn 8268	. . . . . 6 ⊢ 2o ∈ ω
3	2	elexi 3515	. . . . 5 ⊢ 2o ∈ V
4		domrefg 8546	. . . . 5 ⊢ (2o ∈ V → 2o ≼ 2o)
5	3	cfpwsdom 10008	. . . . 5 ⊢ (2o ≼ 2o → (ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅)))))
6	3, 4, 5	mp2b 10	. . . 4 ⊢ (ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅))))
7		aleph0 9494	. . . . . 6 ⊢ (ℵ‘∅) = ω
8	7	a1i 11	. . . . 5 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (ℵ‘∅) = ω)
9	7	oveq2i 7169	. . . . . . . . . 10 ⊢ (2o ↑m (ℵ‘∅)) = (2o ↑m ω)
10	9	fveq2i 6675	. . . . . . . . 9 ⊢ (card‘(2o ↑m (ℵ‘∅))) = (card‘(2o ↑m ω))
11	10	eqeq1i 2828	. . . . . . . 8 ⊢ ((card‘(2o ↑m (ℵ‘∅))) = (ℵ‘ω) ↔ (card‘(2o ↑m ω)) = (ℵ‘ω))
12	11	biimpri 230	. . . . . . 7 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (card‘(2o ↑m (ℵ‘∅))) = (ℵ‘ω))
13	12	fveq2d 6676	. . . . . 6 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (cf‘(card‘(2o ↑m (ℵ‘∅)))) = (cf‘(ℵ‘ω)))
14		limom 7597	. . . . . . . 8 ⊢ Lim ω
15		alephsing 9700	. . . . . . . 8 ⊢ (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω))
16	14, 15	ax-mp 5	. . . . . . 7 ⊢ (cf‘(ℵ‘ω)) = (cf‘ω)
17		cfom 9688	. . . . . . 7 ⊢ (cf‘ω) = ω
18	16, 17	eqtri 2846	. . . . . 6 ⊢ (cf‘(ℵ‘ω)) = ω
19	13, 18	syl6eq 2874	. . . . 5 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (cf‘(card‘(2o ↑m (ℵ‘∅)))) = ω)
20	8, 19	breq12d 5081	. . . 4 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → ((ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅)))) ↔ ω ≺ ω))
21	6, 20	mpbii 235	. . 3 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → ω ≺ ω)
22	21	necon3bi 3044	. 2 ⊢ (¬ ω ≺ ω → (card‘(2o ↑m ω)) ≠ (ℵ‘ω))
23	1, 22	ax-mp 5	1 ⊢ (card‘(2o ↑m ω)) ≠ (ℵ‘ω)

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  Vcvv 3496  ∅c0 4293   class class class wbr 5068  Lim wlim 6194  ‘cfv 6357  (class class class)co 7158  ωcom 7582  2_oc2o 8098   ↑_m cmap 8408   ≼ cdom 8509   ≺ csdm 8510  cardccrd 9366  ℵcale 9367  cfccf 9368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106  ax-ac2 9887

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-smo 7985  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-oi 8976  df-har 9024  df-card 9370  df-aleph 9371  df-cf 9372  df-acn 9373  df-ac 9544

This theorem is referenced by: (None)