Theorem alephom 10697

Description: From canth2 9177, 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 10681 (in the form of cfpwsdom 10696), (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 9161	. 2 ⊢ ¬ ω ≺ ω
2		2onn 8679	. . . . . 6 ⊢ 2o ∈ ω
3	2	elexi 3476	. . . . 5 ⊢ 2o ∈ V
4		domrefg 9042	. . . . 5 ⊢ (2o ∈ V → 2o ≼ 2o)
5	3	cfpwsdom 10696	. . . . 5 ⊢ (2o ≼ 2o → (ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅)))))
6	3, 4, 5	mp2b 10	. . . 4 ⊢ (ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅))))
7		aleph0 10172	. . . . . 6 ⊢ (ℵ‘∅) = ω
8	7	a1i 11	. . . . 5 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (ℵ‘∅) = ω)
9	7	oveq2i 7454	. . . . . . . . . 10 ⊢ (2o ↑m (ℵ‘∅)) = (2o ↑m ω)
10	9	fveq2i 6911	. . . . . . . . 9 ⊢ (card‘(2o ↑m (ℵ‘∅))) = (card‘(2o ↑m ω))
11	10	eqeq1i 2768	. . . . . . . 8 ⊢ ((card‘(2o ↑m (ℵ‘∅))) = (ℵ‘ω) ↔ (card‘(2o ↑m ω)) = (ℵ‘ω))
12	11	biimpri 231	. . . . . . 7 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (card‘(2o ↑m (ℵ‘∅))) = (ℵ‘ω))
13	12	fveq2d 6912	. . . . . 6 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (cf‘(card‘(2o ↑m (ℵ‘∅)))) = (cf‘(ℵ‘ω)))
14		limom 7917	. . . . . . . 8 ⊢ Lim ω
15		alephsing 10381	. . . . . . . 8 ⊢ (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω))
16	14, 15	ax-mp 5	. . . . . . 7 ⊢ (cf‘(ℵ‘ω)) = (cf‘ω)
17		cfom 10369	. . . . . . 7 ⊢ (cf‘ω) = ω
18	16, 17	eqtri 2786	. . . . . 6 ⊢ (cf‘(ℵ‘ω)) = ω
19	13, 18	eqtrdi 2814	. . . . 5 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (cf‘(card‘(2o ↑m (ℵ‘∅)))) = ω)
20	8, 19	breq12d 5119	. . . 4 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → ((ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅)))) ↔ ω ≺ ω))
21	6, 20	mpbii 236	. . 3 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → ω ≺ ω)
22	21	necon3bi 2985	. 2 ⊢ (¬ ω ≺ ω → (card‘(2o ↑m ω)) ≠ (ℵ‘ω))
23	1, 22	ax-mp 5	1 ⊢ (card‘(2o ↑m ω)) ≠ (ℵ‘ω)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454  ∅c0 4281   class class class wbr 5106  Lim wlim 6381  ‘cfv 6558  (class class class)co 7443  ωcom 7901  2_oc2o 8498   ↑_m cmap 8875   ≼ cdom 8999   ≺ csdm 9000  cardccrd 10043  ℵcale 10044  cfccf 10045

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669  ax-ac2 10568

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-smo 8382  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-er 8745  df-map 8877  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-oi 9531  df-har 9578  df-card 10047  df-aleph 10048  df-cf 10049  df-acn 10050  df-ac 10222

This theorem is used by: (None)