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Mirrors > Home > MPE Home > Th. List > alephom | Structured version Visualization version GIF version |
Description: From canth2 9132, 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 10566 (in the form of cfpwsdom 10581), (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 9116 | . 2 ⊢ ¬ ω ≺ ω | |
2 | 2onn 8643 | . . . . . 6 ⊢ 2o ∈ ω | |
3 | 2 | elexi 3493 | . . . . 5 ⊢ 2o ∈ V |
4 | domrefg 8985 | . . . . 5 ⊢ (2o ∈ V → 2o ≼ 2o) | |
5 | 3 | cfpwsdom 10581 | . . . . 5 ⊢ (2o ≼ 2o → (ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅))))) |
6 | 3, 4, 5 | mp2b 10 | . . . 4 ⊢ (ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅)))) |
7 | aleph0 10063 | . . . . . 6 ⊢ (ℵ‘∅) = ω | |
8 | 7 | a1i 11 | . . . . 5 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (ℵ‘∅) = ω) |
9 | 7 | oveq2i 7422 | . . . . . . . . . 10 ⊢ (2o ↑m (ℵ‘∅)) = (2o ↑m ω) |
10 | 9 | fveq2i 6894 | . . . . . . . . 9 ⊢ (card‘(2o ↑m (ℵ‘∅))) = (card‘(2o ↑m ω)) |
11 | 10 | eqeq1i 2737 | . . . . . . . 8 ⊢ ((card‘(2o ↑m (ℵ‘∅))) = (ℵ‘ω) ↔ (card‘(2o ↑m ω)) = (ℵ‘ω)) |
12 | 11 | biimpri 227 | . . . . . . 7 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (card‘(2o ↑m (ℵ‘∅))) = (ℵ‘ω)) |
13 | 12 | fveq2d 6895 | . . . . . 6 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (cf‘(card‘(2o ↑m (ℵ‘∅)))) = (cf‘(ℵ‘ω))) |
14 | limom 7873 | . . . . . . . 8 ⊢ Lim ω | |
15 | alephsing 10273 | . . . . . . . 8 ⊢ (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω)) | |
16 | 14, 15 | ax-mp 5 | . . . . . . 7 ⊢ (cf‘(ℵ‘ω)) = (cf‘ω) |
17 | cfom 10261 | . . . . . . 7 ⊢ (cf‘ω) = ω | |
18 | 16, 17 | eqtri 2760 | . . . . . 6 ⊢ (cf‘(ℵ‘ω)) = ω |
19 | 13, 18 | eqtrdi 2788 | . . . . 5 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (cf‘(card‘(2o ↑m (ℵ‘∅)))) = ω) |
20 | 8, 19 | breq12d 5161 | . . . 4 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → ((ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅)))) ↔ ω ≺ ω)) |
21 | 6, 20 | mpbii 232 | . . 3 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → ω ≺ ω) |
22 | 21 | necon3bi 2967 | . 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 1541 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 ∅c0 4322 class class class wbr 5148 Lim wlim 6365 ‘cfv 6543 (class class class)co 7411 ωcom 7857 2oc2o 8462 ↑m cmap 8822 ≼ cdom 8939 ≺ csdm 8940 cardccrd 9932 ℵcale 9933 cfccf 9934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-ac2 10460 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-smo 8348 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-oi 9507 df-har 9554 df-card 9936 df-aleph 9937 df-cf 9938 df-acn 9939 df-ac 10113 |
This theorem is referenced by: (None) |
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