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Mirrors > Home > MPE Home > Th. List > alephom | Structured version Visualization version GIF version |
Description: From canth2 8821, 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 10207 (in the form of cfpwsdom 10222), (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.) |
Ref | Expression |
---|---|
alephom | ⊢ (card‘(2o ↑m ω)) ≠ (ℵ‘ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomirr 8805 | . 2 ⊢ ¬ ω ≺ ω | |
2 | 2onn 8388 | . . . . . 6 ⊢ 2o ∈ ω | |
3 | 2 | elexi 3439 | . . . . 5 ⊢ 2o ∈ V |
4 | domrefg 8685 | . . . . 5 ⊢ (2o ∈ V → 2o ≼ 2o) | |
5 | 3 | cfpwsdom 10222 | . . . . 5 ⊢ (2o ≼ 2o → (ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅))))) |
6 | 3, 4, 5 | mp2b 10 | . . . 4 ⊢ (ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅)))) |
7 | aleph0 9704 | . . . . . 6 ⊢ (ℵ‘∅) = ω | |
8 | 7 | a1i 11 | . . . . 5 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (ℵ‘∅) = ω) |
9 | 7 | oveq2i 7242 | . . . . . . . . . 10 ⊢ (2o ↑m (ℵ‘∅)) = (2o ↑m ω) |
10 | 9 | fveq2i 6738 | . . . . . . . . 9 ⊢ (card‘(2o ↑m (ℵ‘∅))) = (card‘(2o ↑m ω)) |
11 | 10 | eqeq1i 2743 | . . . . . . . 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 6739 | . . . . . 6 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (cf‘(card‘(2o ↑m (ℵ‘∅)))) = (cf‘(ℵ‘ω))) |
14 | limom 7678 | . . . . . . . 8 ⊢ Lim ω | |
15 | alephsing 9914 | . . . . . . . 8 ⊢ (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω)) | |
16 | 14, 15 | ax-mp 5 | . . . . . . 7 ⊢ (cf‘(ℵ‘ω)) = (cf‘ω) |
17 | cfom 9902 | . . . . . . 7 ⊢ (cf‘ω) = ω | |
18 | 16, 17 | eqtri 2766 | . . . . . 6 ⊢ (cf‘(ℵ‘ω)) = ω |
19 | 13, 18 | eqtrdi 2795 | . . . . 5 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (cf‘(card‘(2o ↑m (ℵ‘∅)))) = ω) |
20 | 8, 19 | breq12d 5080 | . . . 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 2968 | . 2 ⊢ (¬ ω ≺ ω → (card‘(2o ↑m ω)) ≠ (ℵ‘ω)) |
23 | 1, 22 | ax-mp 5 | 1 ⊢ (card‘(2o ↑m ω)) ≠ (ℵ‘ω) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1543 ∈ wcel 2111 ≠ wne 2941 Vcvv 3420 ∅c0 4251 class class class wbr 5067 Lim wlim 6231 ‘cfv 6397 (class class class)co 7231 ωcom 7662 2oc2o 8216 ↑m cmap 8528 ≼ cdom 8644 ≺ csdm 8645 cardccrd 9575 ℵcale 9576 cfccf 9577 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-inf2 9280 ax-ac2 10101 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-int 4874 df-iun 4920 df-iin 4921 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-se 5524 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-isom 6406 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-1st 7779 df-2nd 7780 df-wrecs 8067 df-smo 8103 df-recs 8128 df-rdg 8166 df-1o 8222 df-2o 8223 df-er 8411 df-map 8530 df-ixp 8599 df-en 8647 df-dom 8648 df-sdom 8649 df-fin 8650 df-oi 9150 df-har 9197 df-card 9579 df-aleph 9580 df-cf 9581 df-acn 9582 df-ac 9754 |
This theorem is referenced by: (None) |
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