Theorem alephom 10654

Description: From canth2 9196, 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 10638 (in the form of cfpwsdom 10653), (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 9180	. 2 ⊢ ¬ ω ≺ ω
2		2onn 8698	. . . . . 6 ⊢ 2o ∈ ω
3	2	elexi 3511	. . . . 5 ⊢ 2o ∈ V
4		domrefg 9047	. . . . 5 ⊢ (2o ∈ V → 2o ≼ 2o)
5	3	cfpwsdom 10653	. . . . 5 ⊢ (2o ≼ 2o → (ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅)))))
6	3, 4, 5	mp2b 10	. . . 4 ⊢ (ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅))))
7		aleph0 10135	. . . . . 6 ⊢ (ℵ‘∅) = ω
8	7	a1i 11	. . . . 5 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (ℵ‘∅) = ω)
9	7	oveq2i 7459	. . . . . . . . . 10 ⊢ (2o ↑m (ℵ‘∅)) = (2o ↑m ω)
10	9	fveq2i 6923	. . . . . . . . 9 ⊢ (card‘(2o ↑m (ℵ‘∅))) = (card‘(2o ↑m ω))
11	10	eqeq1i 2745	. . . . . . . 8 ⊢ ((card‘(2o ↑m (ℵ‘∅))) = (ℵ‘ω) ↔ (card‘(2o ↑m ω)) = (ℵ‘ω))
12	11	biimpri 228	. . . . . . 7 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (card‘(2o ↑m (ℵ‘∅))) = (ℵ‘ω))
13	12	fveq2d 6924	. . . . . 6 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (cf‘(card‘(2o ↑m (ℵ‘∅)))) = (cf‘(ℵ‘ω)))
14		limom 7919	. . . . . . . 8 ⊢ Lim ω
15		alephsing 10345	. . . . . . . 8 ⊢ (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω))
16	14, 15	ax-mp 5	. . . . . . 7 ⊢ (cf‘(ℵ‘ω)) = (cf‘ω)
17		cfom 10333	. . . . . . 7 ⊢ (cf‘ω) = ω
18	16, 17	eqtri 2768	. . . . . 6 ⊢ (cf‘(ℵ‘ω)) = ω
19	13, 18	eqtrdi 2796	. . . . 5 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (cf‘(card‘(2o ↑m (ℵ‘∅)))) = ω)
20	8, 19	breq12d 5179	. . . 4 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → ((ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅)))) ↔ ω ≺ ω))
21	6, 20	mpbii 233	. . 3 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → ω ≺ ω)
22	21	necon3bi 2973	. 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 2108   ≠ wne 2946  Vcvv 3488  ∅c0 4352   class class class wbr 5166  Lim wlim 6396  ‘cfv 6573  (class class class)co 7448  ωcom 7903  2_oc2o 8516   ↑_m cmap 8884   ≼ cdom 9001   ≺ csdm 9002  cardccrd 10004  ℵcale 10005  cfccf 10006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-ac2 10532

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-smo 8402  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-oi 9579  df-har 9626  df-card 10008  df-aleph 10009  df-cf 10010  df-acn 10011  df-ac 10185

This theorem is referenced by: (None)