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Mirrors > Home > MPE Home > Th. List > alephom | Structured version Visualization version GIF version |
Description: From canth2 9130, 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 10564 (in the form of cfpwsdom 10579), (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 9114 | . 2 ⊢ ¬ ω ≺ ω | |
2 | 2onn 8641 | . . . . . 6 ⊢ 2o ∈ ω | |
3 | 2 | elexi 3494 | . . . . 5 ⊢ 2o ∈ V |
4 | domrefg 8983 | . . . . 5 ⊢ (2o ∈ V → 2o ≼ 2o) | |
5 | 3 | cfpwsdom 10579 | . . . . 5 ⊢ (2o ≼ 2o → (ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅))))) |
6 | 3, 4, 5 | mp2b 10 | . . . 4 ⊢ (ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅)))) |
7 | aleph0 10061 | . . . . . 6 ⊢ (ℵ‘∅) = ω | |
8 | 7 | a1i 11 | . . . . 5 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (ℵ‘∅) = ω) |
9 | 7 | oveq2i 7420 | . . . . . . . . . 10 ⊢ (2o ↑m (ℵ‘∅)) = (2o ↑m ω) |
10 | 9 | fveq2i 6895 | . . . . . . . . 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 6896 | . . . . . 6 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (cf‘(card‘(2o ↑m (ℵ‘∅)))) = (cf‘(ℵ‘ω))) |
14 | limom 7871 | . . . . . . . 8 ⊢ Lim ω | |
15 | alephsing 10271 | . . . . . . . 8 ⊢ (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω)) | |
16 | 14, 15 | ax-mp 5 | . . . . . . 7 ⊢ (cf‘(ℵ‘ω)) = (cf‘ω) |
17 | cfom 10259 | . . . . . . 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 5162 | . . . 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 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 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ∅c0 4323 class class class wbr 5149 Lim wlim 6366 ‘cfv 6544 (class class class)co 7409 ωcom 7855 2oc2o 8460 ↑m cmap 8820 ≼ cdom 8937 ≺ csdm 8938 cardccrd 9930 ℵcale 9931 cfccf 9932 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-ac2 10458 |
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 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-smo 8346 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-oi 9505 df-har 9552 df-card 9934 df-aleph 9935 df-cf 9936 df-acn 9937 df-ac 10111 |
This theorem is referenced by: (None) |
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