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Mirrors > Home > MPE Home > Th. List > aleph1 | Structured version Visualization version GIF version |
Description: The set exponentiation of 2 to the aleph-zero has cardinality of at least aleph-one. (If we were to assume the Continuum Hypothesis, their cardinalities would be the same.) (Contributed by NM, 7-Jul-2004.) |
Ref | Expression |
---|---|
aleph1 | ⊢ (ℵ‘1o) ≼ (2o ↑m (ℵ‘∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1o 8496 | . . 3 ⊢ 1o = suc ∅ | |
2 | 1 | fveq2i 6904 | . 2 ⊢ (ℵ‘1o) = (ℵ‘suc ∅) |
3 | alephsucpw 10613 | . . 3 ⊢ (ℵ‘suc ∅) ≼ 𝒫 (ℵ‘∅) | |
4 | fvex 6914 | . . . . 5 ⊢ (ℵ‘∅) ∈ V | |
5 | 4 | pw2en 9117 | . . . 4 ⊢ 𝒫 (ℵ‘∅) ≈ (2o ↑m (ℵ‘∅)) |
6 | domen2 9158 | . . . 4 ⊢ (𝒫 (ℵ‘∅) ≈ (2o ↑m (ℵ‘∅)) → ((ℵ‘suc ∅) ≼ 𝒫 (ℵ‘∅) ↔ (ℵ‘suc ∅) ≼ (2o ↑m (ℵ‘∅)))) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ ((ℵ‘suc ∅) ≼ 𝒫 (ℵ‘∅) ↔ (ℵ‘suc ∅) ≼ (2o ↑m (ℵ‘∅))) |
8 | 3, 7 | mpbi 229 | . 2 ⊢ (ℵ‘suc ∅) ≼ (2o ↑m (ℵ‘∅)) |
9 | 2, 8 | eqbrtri 5174 | 1 ⊢ (ℵ‘1o) ≼ (2o ↑m (ℵ‘∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∅c0 4325 𝒫 cpw 4607 class class class wbr 5153 suc csuc 6378 ‘cfv 6554 (class class class)co 7424 1oc1o 8489 2oc2o 8490 ↑m cmap 8855 ≈ cen 8971 ≼ cdom 8972 ℵcale 9979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-ac2 10506 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-oi 9553 df-har 9600 df-card 9982 df-aleph 9983 df-ac 10159 |
This theorem is referenced by: (None) |
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