Theorem alephom 9286

Description: From canth2 7998, 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 9270 (in the form of cfpwsdom 9285), (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 7982	. 2 ⊢ ¬ ω ≺ ω
2		2onn 7607	. . . . . 6 ⊢ 2𝑜 ∈ ω
3	2	elexi 3186	. . . . 5 ⊢ 2𝑜 ∈ V
4		domrefg 7876	. . . . 5 ⊢ (2𝑜 ∈ V → 2𝑜 ≼ 2𝑜)
5	3	cfpwsdom 9285	. . . . 5 ⊢ (2𝑜 ≼ 2𝑜 → (ℵ‘∅) ≺ (cf‘(card‘(2𝑜 ↑𝑚 (ℵ‘∅)))))
6	3, 4, 5	mp2b 10	. . . 4 ⊢ (ℵ‘∅) ≺ (cf‘(card‘(2𝑜 ↑𝑚 (ℵ‘∅))))
7		aleph0 8772	. . . . . 6 ⊢ (ℵ‘∅) = ω
8	7	a1i 11	. . . . 5 ⊢ ((card‘(2𝑜 ↑𝑚 ω)) = (ℵ‘ω) → (ℵ‘∅) = ω)
9	7	oveq2i 6560	. . . . . . . . . 10 ⊢ (2𝑜 ↑𝑚 (ℵ‘∅)) = (2𝑜 ↑𝑚 ω)
10	9	fveq2i 6106	. . . . . . . . 9 ⊢ (card‘(2𝑜 ↑𝑚 (ℵ‘∅))) = (card‘(2𝑜 ↑𝑚 ω))
11	10	eqeq1i 2615	. . . . . . . 8 ⊢ ((card‘(2𝑜 ↑𝑚 (ℵ‘∅))) = (ℵ‘ω) ↔ (card‘(2𝑜 ↑𝑚 ω)) = (ℵ‘ω))
12	11	biimpri 217	. . . . . . 7 ⊢ ((card‘(2𝑜 ↑𝑚 ω)) = (ℵ‘ω) → (card‘(2𝑜 ↑𝑚 (ℵ‘∅))) = (ℵ‘ω))
13	12	fveq2d 6107	. . . . . 6 ⊢ ((card‘(2𝑜 ↑𝑚 ω)) = (ℵ‘ω) → (cf‘(card‘(2𝑜 ↑𝑚 (ℵ‘∅)))) = (cf‘(ℵ‘ω)))
14		limom 6972	. . . . . . . 8 ⊢ Lim ω
15		alephsing 8981	. . . . . . . 8 ⊢ (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω))
16	14, 15	ax-mp 5	. . . . . . 7 ⊢ (cf‘(ℵ‘ω)) = (cf‘ω)
17		cfom 8969	. . . . . . 7 ⊢ (cf‘ω) = ω
18	16, 17	eqtri 2632	. . . . . 6 ⊢ (cf‘(ℵ‘ω)) = ω
19	13, 18	syl6eq 2660	. . . . 5 ⊢ ((card‘(2𝑜 ↑𝑚 ω)) = (ℵ‘ω) → (cf‘(card‘(2𝑜 ↑𝑚 (ℵ‘∅)))) = ω)
20	8, 19	breq12d 4596	. . . 4 ⊢ ((card‘(2𝑜 ↑𝑚 ω)) = (ℵ‘ω) → ((ℵ‘∅) ≺ (cf‘(card‘(2𝑜 ↑𝑚 (ℵ‘∅)))) ↔ ω ≺ ω))
21	6, 20	mpbii 222	. . 3 ⊢ ((card‘(2𝑜 ↑𝑚 ω)) = (ℵ‘ω) → ω ≺ ω)
22	21	necon3bi 2808	. 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 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ∅c0 3874   class class class wbr 4583  Lim wlim 5641  ‘cfv 5804  (class class class)co 6549  ωcom 6957  2_𝑜c2o 7441   ↑_𝑚 cmap 7744   ≼ cdom 7839   ≺ csdm 7840  cardccrd 8644  ℵcale 8645  cfccf 8646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-ac2 9168

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-smo 7330  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-har 8346  df-card 8648  df-aleph 8649  df-cf 8650  df-acn 8651  df-ac 8822

This theorem is referenced by: (None)