Theorem alephom 9407

Description: From canth2 8113, 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 9391 (in the form of cfpwsdom 9406), (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 8097	. 2 ⊢ ¬ ω ≺ ω
2		2onn 7720	. . . . . 6 ⊢ 2𝑜 ∈ ω
3	2	elexi 3213	. . . . 5 ⊢ 2𝑜 ∈ V
4		domrefg 7990	. . . . 5 ⊢ (2𝑜 ∈ V → 2𝑜 ≼ 2𝑜)
5	3	cfpwsdom 9406	. . . . 5 ⊢ (2𝑜 ≼ 2𝑜 → (ℵ‘∅) ≺ (cf‘(card‘(2𝑜 ↑𝑚 (ℵ‘∅)))))
6	3, 4, 5	mp2b 10	. . . 4 ⊢ (ℵ‘∅) ≺ (cf‘(card‘(2𝑜 ↑𝑚 (ℵ‘∅))))
7		aleph0 8889	. . . . . 6 ⊢ (ℵ‘∅) = ω
8	7	a1i 11	. . . . 5 ⊢ ((card‘(2𝑜 ↑𝑚 ω)) = (ℵ‘ω) → (ℵ‘∅) = ω)
9	7	oveq2i 6661	. . . . . . . . . 10 ⊢ (2𝑜 ↑𝑚 (ℵ‘∅)) = (2𝑜 ↑𝑚 ω)
10	9	fveq2i 6194	. . . . . . . . 9 ⊢ (card‘(2𝑜 ↑𝑚 (ℵ‘∅))) = (card‘(2𝑜 ↑𝑚 ω))
11	10	eqeq1i 2627	. . . . . . . 8 ⊢ ((card‘(2𝑜 ↑𝑚 (ℵ‘∅))) = (ℵ‘ω) ↔ (card‘(2𝑜 ↑𝑚 ω)) = (ℵ‘ω))
12	11	biimpri 218	. . . . . . 7 ⊢ ((card‘(2𝑜 ↑𝑚 ω)) = (ℵ‘ω) → (card‘(2𝑜 ↑𝑚 (ℵ‘∅))) = (ℵ‘ω))
13	12	fveq2d 6195	. . . . . 6 ⊢ ((card‘(2𝑜 ↑𝑚 ω)) = (ℵ‘ω) → (cf‘(card‘(2𝑜 ↑𝑚 (ℵ‘∅)))) = (cf‘(ℵ‘ω)))
14		limom 7080	. . . . . . . 8 ⊢ Lim ω
15		alephsing 9098	. . . . . . . 8 ⊢ (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω))
16	14, 15	ax-mp 5	. . . . . . 7 ⊢ (cf‘(ℵ‘ω)) = (cf‘ω)
17		cfom 9086	. . . . . . 7 ⊢ (cf‘ω) = ω
18	16, 17	eqtri 2644	. . . . . 6 ⊢ (cf‘(ℵ‘ω)) = ω
19	13, 18	syl6eq 2672	. . . . 5 ⊢ ((card‘(2𝑜 ↑𝑚 ω)) = (ℵ‘ω) → (cf‘(card‘(2𝑜 ↑𝑚 (ℵ‘∅)))) = ω)
20	8, 19	breq12d 4666	. . . 4 ⊢ ((card‘(2𝑜 ↑𝑚 ω)) = (ℵ‘ω) → ((ℵ‘∅) ≺ (cf‘(card‘(2𝑜 ↑𝑚 (ℵ‘∅)))) ↔ ω ≺ ω))
21	6, 20	mpbii 223	. . 3 ⊢ ((card‘(2𝑜 ↑𝑚 ω)) = (ℵ‘ω) → ω ≺ ω)
22	21	necon3bi 2820	. 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 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200  ∅c0 3915   class class class wbr 4653  Lim wlim 5724  ‘cfv 5888  (class class class)co 6650  ωcom 7065  2_𝑜c2o 7554   ↑_𝑚 cmap 7857   ≼ cdom 7953   ≺ csdm 7954  cardccrd 8761  ℵcale 8762  cfccf 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-smo 7443  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766  df-cf 8767  df-acn 8768  df-ac 8939

This theorem is referenced by: (None)