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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 7447 | . . 3 ⊢ 1𝑜 = suc ∅ | |
2 | 1 | fveq2i 6106 | . 2 ⊢ (ℵ‘1𝑜) = (ℵ‘suc ∅) |
3 | alephsucpw 9271 | . . 3 ⊢ (ℵ‘suc ∅) ≼ 𝒫 (ℵ‘∅) | |
4 | fvex 6113 | . . . . 5 ⊢ (ℵ‘∅) ∈ V | |
5 | 4 | pw2en 7952 | . . . 4 ⊢ 𝒫 (ℵ‘∅) ≈ (2𝑜 ↑𝑚 (ℵ‘∅)) |
6 | domen2 7988 | . . . 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 219 | . 2 ⊢ (ℵ‘suc ∅) ≼ (2𝑜 ↑𝑚 (ℵ‘∅)) |
9 | 2, 8 | eqbrtri 4604 | 1 ⊢ (ℵ‘1𝑜) ≼ (2𝑜 ↑𝑚 (ℵ‘∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∅c0 3874 𝒫 cpw 4108 class class class wbr 4583 suc csuc 5642 ‘cfv 5804 (class class class)co 6549 1𝑜c1o 7440 2𝑜c2o 7441 ↑𝑚 cmap 7744 ≈ cen 7838 ≼ cdom 7839 ℵcale 8645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-ac2 9168 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-oi 8298 df-har 8346 df-card 8648 df-aleph 8649 df-ac 8822 |
This theorem is referenced by: (None) |
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