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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 14811, which only asserts that ℝ is uncountable, i.e. has a cardinality larger than ω. The main proof is in two parts, rpnnen1 11696 and rpnnen2 14794, each showing an injection in one direction, and this last part uses sbth 7965 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 10903 | . . . 4 ⊢ ℕ ∈ V | |
2 | qex 11676 | . . . 4 ⊢ ℚ ∈ V | |
3 | 1, 2 | rpnnen1 11696 | . . 3 ⊢ ℝ ≼ (ℚ ↑𝑚 ℕ) |
4 | qnnen 14781 | . . . . . . 7 ⊢ ℚ ≈ ℕ | |
5 | 1 | canth2 7998 | . . . . . . 7 ⊢ ℕ ≺ 𝒫 ℕ |
6 | ensdomtr 7981 | . . . . . . 7 ⊢ ((ℚ ≈ ℕ ∧ ℕ ≺ 𝒫 ℕ) → ℚ ≺ 𝒫 ℕ) | |
7 | 4, 5, 6 | mp2an 704 | . . . . . 6 ⊢ ℚ ≺ 𝒫 ℕ |
8 | sdomdom 7869 | . . . . . 6 ⊢ (ℚ ≺ 𝒫 ℕ → ℚ ≼ 𝒫 ℕ) | |
9 | mapdom1 8010 | . . . . . 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 7952 | . . . . . 6 ⊢ 𝒫 ℕ ≈ (2𝑜 ↑𝑚 ℕ) |
12 | 1 | enref 7874 | . . . . . 6 ⊢ ℕ ≈ ℕ |
13 | mapen 8009 | . . . . . 6 ⊢ ((𝒫 ℕ ≈ (2𝑜 ↑𝑚 ℕ) ∧ ℕ ≈ ℕ) → (𝒫 ℕ ↑𝑚 ℕ) ≈ ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ)) | |
14 | 11, 12, 13 | mp2an 704 | . . . . 5 ⊢ (𝒫 ℕ ↑𝑚 ℕ) ≈ ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ) |
15 | domentr 7901 | . . . . 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 704 | . . . 4 ⊢ (ℚ ↑𝑚 ℕ) ≼ ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ) |
17 | 2onn 7607 | . . . . . . 7 ⊢ 2𝑜 ∈ ω | |
18 | mapxpen 8011 | . . . . . . 7 ⊢ ((2𝑜 ∈ ω ∧ ℕ ∈ V ∧ ℕ ∈ V) → ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ) ≈ (2𝑜 ↑𝑚 (ℕ × ℕ))) | |
19 | 17, 1, 1, 18 | mp3an 1416 | . . . . . 6 ⊢ ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ) ≈ (2𝑜 ↑𝑚 (ℕ × ℕ)) |
20 | 17 | elexi 3186 | . . . . . . . 8 ⊢ 2𝑜 ∈ V |
21 | 20 | enref 7874 | . . . . . . 7 ⊢ 2𝑜 ≈ 2𝑜 |
22 | xpnnen 14778 | . . . . . . 7 ⊢ (ℕ × ℕ) ≈ ℕ | |
23 | mapen 8009 | . . . . . . 7 ⊢ ((2𝑜 ≈ 2𝑜 ∧ (ℕ × ℕ) ≈ ℕ) → (2𝑜 ↑𝑚 (ℕ × ℕ)) ≈ (2𝑜 ↑𝑚 ℕ)) | |
24 | 21, 22, 23 | mp2an 704 | . . . . . 6 ⊢ (2𝑜 ↑𝑚 (ℕ × ℕ)) ≈ (2𝑜 ↑𝑚 ℕ) |
25 | 19, 24 | entri 7896 | . . . . 5 ⊢ ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ) ≈ (2𝑜 ↑𝑚 ℕ) |
26 | 25, 11 | entr4i 7899 | . . . 4 ⊢ ((2𝑜 ↑𝑚 ℕ) ↑𝑚 ℕ) ≈ 𝒫 ℕ |
27 | domentr 7901 | . . . 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 704 | . . 3 ⊢ (ℚ ↑𝑚 ℕ) ≼ 𝒫 ℕ |
29 | domtr 7895 | . . 3 ⊢ ((ℝ ≼ (ℚ ↑𝑚 ℕ) ∧ (ℚ ↑𝑚 ℕ) ≼ 𝒫 ℕ) → ℝ ≼ 𝒫 ℕ) | |
30 | 3, 28, 29 | mp2an 704 | . 2 ⊢ ℝ ≼ 𝒫 ℕ |
31 | rpnnen2 14794 | . . 3 ⊢ 𝒫 ℕ ≼ (0[,]1) | |
32 | reex 9906 | . . . 4 ⊢ ℝ ∈ V | |
33 | unitssre 12190 | . . . 4 ⊢ (0[,]1) ⊆ ℝ | |
34 | ssdomg 7887 | . . . 4 ⊢ (ℝ ∈ V → ((0[,]1) ⊆ ℝ → (0[,]1) ≼ ℝ)) | |
35 | 32, 33, 34 | mp2 9 | . . 3 ⊢ (0[,]1) ≼ ℝ |
36 | domtr 7895 | . . 3 ⊢ ((𝒫 ℕ ≼ (0[,]1) ∧ (0[,]1) ≼ ℝ) → 𝒫 ℕ ≼ ℝ) | |
37 | 31, 35, 36 | mp2an 704 | . 2 ⊢ 𝒫 ℕ ≼ ℝ |
38 | sbth 7965 | . 2 ⊢ ((ℝ ≼ 𝒫 ℕ ∧ 𝒫 ℕ ≼ ℝ) → ℝ ≈ 𝒫 ℕ) | |
39 | 30, 37, 38 | mp2an 704 | 1 ⊢ ℝ ≈ 𝒫 ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 class class class wbr 4583 × cxp 5036 (class class class)co 6549 ωcom 6957 2𝑜c2o 7441 ↑𝑚 cmap 7744 ≈ cen 7838 ≼ cdom 7839 ≺ csdm 7840 ℝcr 9814 0cc0 9815 1c1 9816 ℕcn 10897 ℚcq 11664 [,]cicc 12049 |
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-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 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-nel 2783 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-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 |
This theorem is referenced by: rexpen 14796 cpnnen 14797 rucALT 14798 cnso 14815 2ndcredom 21063 opnreen 22442 |
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