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Mirrors > Home > MPE Home > Th. List > alephom | Structured version Visualization version GIF version |
Description: From canth2 9080, 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 10513 (in the form of cfpwsdom 10528), (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 9064 | . 2 ⊢ ¬ ω ≺ ω | |
2 | 2onn 8592 | . . . . . 6 ⊢ 2o ∈ ω | |
3 | 2 | elexi 3466 | . . . . 5 ⊢ 2o ∈ V |
4 | domrefg 8933 | . . . . 5 ⊢ (2o ∈ V → 2o ≼ 2o) | |
5 | 3 | cfpwsdom 10528 | . . . . 5 ⊢ (2o ≼ 2o → (ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅))))) |
6 | 3, 4, 5 | mp2b 10 | . . . 4 ⊢ (ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅)))) |
7 | aleph0 10010 | . . . . . 6 ⊢ (ℵ‘∅) = ω | |
8 | 7 | a1i 11 | . . . . 5 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (ℵ‘∅) = ω) |
9 | 7 | oveq2i 7372 | . . . . . . . . . 10 ⊢ (2o ↑m (ℵ‘∅)) = (2o ↑m ω) |
10 | 9 | fveq2i 6849 | . . . . . . . . 9 ⊢ (card‘(2o ↑m (ℵ‘∅))) = (card‘(2o ↑m ω)) |
11 | 10 | eqeq1i 2738 | . . . . . . . 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 6850 | . . . . . 6 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (cf‘(card‘(2o ↑m (ℵ‘∅)))) = (cf‘(ℵ‘ω))) |
14 | limom 7822 | . . . . . . . 8 ⊢ Lim ω | |
15 | alephsing 10220 | . . . . . . . 8 ⊢ (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω)) | |
16 | 14, 15 | ax-mp 5 | . . . . . . 7 ⊢ (cf‘(ℵ‘ω)) = (cf‘ω) |
17 | cfom 10208 | . . . . . . 7 ⊢ (cf‘ω) = ω | |
18 | 16, 17 | eqtri 2761 | . . . . . 6 ⊢ (cf‘(ℵ‘ω)) = ω |
19 | 13, 18 | eqtrdi 2789 | . . . . 5 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (cf‘(card‘(2o ↑m (ℵ‘∅)))) = ω) |
20 | 8, 19 | breq12d 5122 | . . . 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 1542 ∈ wcel 2107 ≠ wne 2940 Vcvv 3447 ∅c0 4286 class class class wbr 5109 Lim wlim 6322 ‘cfv 6500 (class class class)co 7361 ωcom 7806 2oc2o 8410 ↑m cmap 8771 ≼ cdom 8887 ≺ csdm 8888 cardccrd 9879 ℵcale 9880 cfccf 9881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-ac2 10407 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-smo 8296 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-oi 9454 df-har 9501 df-card 9883 df-aleph 9884 df-cf 9885 df-acn 9886 df-ac 10060 |
This theorem is referenced by: (None) |
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