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Mirrors > Home > MPE Home > Th. List > rpnnen | Structured version Visualization version GIF version |
Description: The cardinality of the continuum is the same as the powerset of ω. This is a stronger statement than ruc 15016, which only asserts that ℝ is uncountable, i.e. has a cardinality larger than ω. The main proof is in two parts, rpnnen1 11858 and rpnnen2 14999, each showing an injection in one direction, and this last part uses sbth 8121 to prove that the sets are equinumerous. By constructing explicit injections, we avoid the use of AC. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 23-Aug-2014.) |
Ref | Expression |
---|---|
rpnnen | ⊢ ℝ ≈ 𝒫 ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnex 11064 | . . . 4 ⊢ ℕ ∈ V | |
2 | qex 11838 | . . . 4 ⊢ ℚ ∈ V | |
3 | 1, 2 | rpnnen1 11858 | . . 3 ⊢ ℝ ≼ (ℚ ↑𝑚 ℕ) |
4 | qnnen 14986 | . . . . . . 7 ⊢ ℚ ≈ ℕ | |
5 | 1 | canth2 8154 | . . . . . . 7 ⊢ ℕ ≺ 𝒫 ℕ |
6 | ensdomtr 8137 | . . . . . . 7 ⊢ ((ℚ ≈ ℕ ∧ ℕ ≺ 𝒫 ℕ) → ℚ ≺ 𝒫 ℕ) | |
7 | 4, 5, 6 | mp2an 708 | . . . . . 6 ⊢ ℚ ≺ 𝒫 ℕ |
8 | sdomdom 8025 | . . . . . 6 ⊢ (ℚ ≺ 𝒫 ℕ → ℚ ≼ 𝒫 ℕ) | |
9 | mapdom1 8166 | . . . . . 6 ⊢ (ℚ ≼ 𝒫 ℕ → (ℚ ↑𝑚 ℕ) ≼ (𝒫 ℕ ↑𝑚 ℕ)) | |
10 | 7, 8, 9 | mp2b 10 | . . . . 5 ⊢ (ℚ ↑𝑚 ℕ) ≼ (𝒫 ℕ ↑𝑚 ℕ) |
11 | 1 | pw2en 8108 | . . . . . 6 ⊢ 𝒫 ℕ ≈ (2𝑜 ↑𝑚 ℕ) |
12 | 1 | enref 8030 | . . . . . 6 ⊢ ℕ ≈ ℕ |
13 | mapen 8165 | . . . . . 6 ⊢ ((𝒫 ℕ ≈ (2𝑜 ↑𝑚 ℕ) ∧ ℕ ≈ ℕ) → (𝒫 ℕ ↑𝑚 ℕ) ≈ ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ)) | |
14 | 11, 12, 13 | mp2an 708 | . . . . 5 ⊢ (𝒫 ℕ ↑𝑚 ℕ) ≈ ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ) |
15 | domentr 8056 | . . . . 5 ⊢ (((ℚ ↑𝑚 ℕ) ≼ (𝒫 ℕ ↑𝑚 ℕ) ∧ (𝒫 ℕ ↑𝑚 ℕ) ≈ ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ)) → (ℚ ↑𝑚 ℕ) ≼ ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ)) | |
16 | 10, 14, 15 | mp2an 708 | . . . 4 ⊢ (ℚ ↑𝑚 ℕ) ≼ ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ) |
17 | 2onn 7765 | . . . . . . 7 ⊢ 2𝑜 ∈ ω | |
18 | mapxpen 8167 | . . . . . . 7 ⊢ ((2𝑜 ∈ ω ∧ ℕ ∈ V ∧ ℕ ∈ V) → ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ) ≈ (2𝑜 ↑𝑚 (ℕ × ℕ))) | |
19 | 17, 1, 1, 18 | mp3an 1464 | . . . . . 6 ⊢ ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ) ≈ (2𝑜 ↑𝑚 (ℕ × ℕ)) |
20 | 17 | elexi 3244 | . . . . . . . 8 ⊢ 2𝑜 ∈ V |
21 | 20 | enref 8030 | . . . . . . 7 ⊢ 2𝑜 ≈ 2𝑜 |
22 | xpnnen 14983 | . . . . . . 7 ⊢ (ℕ × ℕ) ≈ ℕ | |
23 | mapen 8165 | . . . . . . 7 ⊢ ((2𝑜 ≈ 2𝑜 ∧ (ℕ × ℕ) ≈ ℕ) → (2𝑜 ↑𝑚 (ℕ × ℕ)) ≈ (2𝑜 ↑𝑚 ℕ)) | |
24 | 21, 22, 23 | mp2an 708 | . . . . . 6 ⊢ (2𝑜 ↑𝑚 (ℕ × ℕ)) ≈ (2𝑜 ↑𝑚 ℕ) |
25 | 19, 24 | entri 8051 | . . . . 5 ⊢ ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ) ≈ (2𝑜 ↑𝑚 ℕ) |
26 | 25, 11 | entr4i 8054 | . . . 4 ⊢ ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ) ≈ 𝒫 ℕ |
27 | domentr 8056 | . . . 4 ⊢ (((ℚ ↑𝑚 ℕ) ≼ ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ) ∧ ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ) ≈ 𝒫 ℕ) → (ℚ ↑𝑚 ℕ) ≼ 𝒫 ℕ) | |
28 | 16, 26, 27 | mp2an 708 | . . 3 ⊢ (ℚ ↑𝑚 ℕ) ≼ 𝒫 ℕ |
29 | domtr 8050 | . . 3 ⊢ ((ℝ ≼ (ℚ ↑𝑚 ℕ) ∧ (ℚ ↑𝑚 ℕ) ≼ 𝒫 ℕ) → ℝ ≼ 𝒫 ℕ) | |
30 | 3, 28, 29 | mp2an 708 | . 2 ⊢ ℝ ≼ 𝒫 ℕ |
31 | rpnnen2 14999 | . . 3 ⊢ 𝒫 ℕ ≼ (0[,]1) | |
32 | reex 10065 | . . . 4 ⊢ ℝ ∈ V | |
33 | unitssre 12357 | . . . 4 ⊢ (0[,]1) ⊆ ℝ | |
34 | ssdomg 8043 | . . . 4 ⊢ (ℝ ∈ V → ((0[,]1) ⊆ ℝ → (0[,]1) ≼ ℝ)) | |
35 | 32, 33, 34 | mp2 9 | . . 3 ⊢ (0[,]1) ≼ ℝ |
36 | domtr 8050 | . . 3 ⊢ ((𝒫 ℕ ≼ (0[,]1) ∧ (0[,]1) ≼ ℝ) → 𝒫 ℕ ≼ ℝ) | |
37 | 31, 35, 36 | mp2an 708 | . 2 ⊢ 𝒫 ℕ ≼ ℝ |
38 | sbth 8121 | . 2 ⊢ ((ℝ ≼ 𝒫 ℕ ∧ 𝒫 ℕ ≼ ℝ) → ℝ ≈ 𝒫 ℕ) | |
39 | 30, 37, 38 | mp2an 708 | 1 ⊢ ℝ ≈ 𝒫 ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2030 Vcvv 3231 ⊆ wss 3607 𝒫 cpw 4191 class class class wbr 4685 × cxp 5141 (class class class)co 6690 ωcom 7107 2𝑜c2o 7599 ↑𝑚 cmap 7899 ≈ cen 7994 ≼ cdom 7995 ≺ csdm 7996 ℝcr 9973 0cc0 9974 1c1 9975 ℕcn 11058 ℚcq 11826 [,]cicc 12216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-omul 7610 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-acn 8806 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-z 11416 df-uz 11726 df-q 11827 df-rp 11871 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 |
This theorem is referenced by: rexpen 15001 cpnnen 15002 rucALT 15003 cnso 15020 2ndcredom 21301 opnreen 22681 |
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