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| Description: From canth2 9171, 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 10610 (in the form of cfpwsdom 10625), (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 9155 | . 2 ⊢ ¬ ω ≺ ω | |
| 2 | 2onn 8681 | . . . . . 6 ⊢ 2o ∈ ω | |
| 3 | 2 | elexi 3502 | . . . . 5 ⊢ 2o ∈ V | 
| 4 | domrefg 9028 | . . . . 5 ⊢ (2o ∈ V → 2o ≼ 2o) | |
| 5 | 3 | cfpwsdom 10625 | . . . . 5 ⊢ (2o ≼ 2o → (ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅))))) | 
| 6 | 3, 4, 5 | mp2b 10 | . . . 4 ⊢ (ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅)))) | 
| 7 | aleph0 10107 | . . . . . 6 ⊢ (ℵ‘∅) = ω | |
| 8 | 7 | a1i 11 | . . . . 5 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (ℵ‘∅) = ω) | 
| 9 | 7 | oveq2i 7443 | . . . . . . . . . 10 ⊢ (2o ↑m (ℵ‘∅)) = (2o ↑m ω) | 
| 10 | 9 | fveq2i 6908 | . . . . . . . . 9 ⊢ (card‘(2o ↑m (ℵ‘∅))) = (card‘(2o ↑m ω)) | 
| 11 | 10 | eqeq1i 2741 | . . . . . . . 8 ⊢ ((card‘(2o ↑m (ℵ‘∅))) = (ℵ‘ω) ↔ (card‘(2o ↑m ω)) = (ℵ‘ω)) | 
| 12 | 11 | biimpri 228 | . . . . . . 7 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (card‘(2o ↑m (ℵ‘∅))) = (ℵ‘ω)) | 
| 13 | 12 | fveq2d 6909 | . . . . . 6 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (cf‘(card‘(2o ↑m (ℵ‘∅)))) = (cf‘(ℵ‘ω))) | 
| 14 | limom 7904 | . . . . . . . 8 ⊢ Lim ω | |
| 15 | alephsing 10317 | . . . . . . . 8 ⊢ (Lim ω → (cf‘(ℵ‘ω)) = (cf‘ω)) | |
| 16 | 14, 15 | ax-mp 5 | . . . . . . 7 ⊢ (cf‘(ℵ‘ω)) = (cf‘ω) | 
| 17 | cfom 10305 | . . . . . . 7 ⊢ (cf‘ω) = ω | |
| 18 | 16, 17 | eqtri 2764 | . . . . . 6 ⊢ (cf‘(ℵ‘ω)) = ω | 
| 19 | 13, 18 | eqtrdi 2792 | . . . . 5 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → (cf‘(card‘(2o ↑m (ℵ‘∅)))) = ω) | 
| 20 | 8, 19 | breq12d 5155 | . . . 4 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → ((ℵ‘∅) ≺ (cf‘(card‘(2o ↑m (ℵ‘∅)))) ↔ ω ≺ ω)) | 
| 21 | 6, 20 | mpbii 233 | . . 3 ⊢ ((card‘(2o ↑m ω)) = (ℵ‘ω) → ω ≺ ω) | 
| 22 | 21 | necon3bi 2966 | . 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 1539 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 ∅c0 4332 class class class wbr 5142 Lim wlim 6384 ‘cfv 6560 (class class class)co 7432 ωcom 7888 2oc2o 8501 ↑m cmap 8867 ≼ cdom 8984 ≺ csdm 8985 cardccrd 9976 ℵcale 9977 cfccf 9978 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-ac2 10504 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-smo 8387 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-oi 9551 df-har 9598 df-card 9980 df-aleph 9981 df-cf 9982 df-acn 9983 df-ac 10157 | 
| This theorem is referenced by: (None) | 
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