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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 15804, which only asserts that ℝ is uncountable, i.e. has a cardinality larger than ω. The main proof is in two parts, rpnnen1 12579 and rpnnen2 15787, each showing an injection in one direction, and this last part uses sbth 8766 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 11836 | . . . 4 ⊢ ℕ ∈ V | |
2 | qex 12557 | . . . 4 ⊢ ℚ ∈ V | |
3 | 1, 2 | rpnnen1 12579 | . . 3 ⊢ ℝ ≼ (ℚ ↑m ℕ) |
4 | qnnen 15774 | . . . . . . 7 ⊢ ℚ ≈ ℕ | |
5 | 1 | canth2 8799 | . . . . . . 7 ⊢ ℕ ≺ 𝒫 ℕ |
6 | ensdomtr 8782 | . . . . . . 7 ⊢ ((ℚ ≈ ℕ ∧ ℕ ≺ 𝒫 ℕ) → ℚ ≺ 𝒫 ℕ) | |
7 | 4, 5, 6 | mp2an 692 | . . . . . 6 ⊢ ℚ ≺ 𝒫 ℕ |
8 | sdomdom 8656 | . . . . . 6 ⊢ (ℚ ≺ 𝒫 ℕ → ℚ ≼ 𝒫 ℕ) | |
9 | mapdom1 8811 | . . . . . 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 8752 | . . . . . 6 ⊢ 𝒫 ℕ ≈ (2o ↑m ℕ) |
12 | 1 | enref 8661 | . . . . . 6 ⊢ ℕ ≈ ℕ |
13 | mapen 8810 | . . . . . 6 ⊢ ((𝒫 ℕ ≈ (2o ↑m ℕ) ∧ ℕ ≈ ℕ) → (𝒫 ℕ ↑m ℕ) ≈ ((2o ↑m ℕ) ↑m ℕ)) | |
14 | 11, 12, 13 | mp2an 692 | . . . . 5 ⊢ (𝒫 ℕ ↑m ℕ) ≈ ((2o ↑m ℕ) ↑m ℕ) |
15 | domentr 8687 | . . . . 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 692 | . . . 4 ⊢ (ℚ ↑m ℕ) ≼ ((2o ↑m ℕ) ↑m ℕ) |
17 | 2onn 8368 | . . . . . . 7 ⊢ 2o ∈ ω | |
18 | mapxpen 8812 | . . . . . . 7 ⊢ ((2o ∈ ω ∧ ℕ ∈ V ∧ ℕ ∈ V) → ((2o ↑m ℕ) ↑m ℕ) ≈ (2o ↑m (ℕ × ℕ))) | |
19 | 17, 1, 1, 18 | mp3an 1463 | . . . . . 6 ⊢ ((2o ↑m ℕ) ↑m ℕ) ≈ (2o ↑m (ℕ × ℕ)) |
20 | 17 | elexi 3427 | . . . . . . . 8 ⊢ 2o ∈ V |
21 | 20 | enref 8661 | . . . . . . 7 ⊢ 2o ≈ 2o |
22 | xpnnen 15772 | . . . . . . 7 ⊢ (ℕ × ℕ) ≈ ℕ | |
23 | mapen 8810 | . . . . . . 7 ⊢ ((2o ≈ 2o ∧ (ℕ × ℕ) ≈ ℕ) → (2o ↑m (ℕ × ℕ)) ≈ (2o ↑m ℕ)) | |
24 | 21, 22, 23 | mp2an 692 | . . . . . 6 ⊢ (2o ↑m (ℕ × ℕ)) ≈ (2o ↑m ℕ) |
25 | 19, 24 | entri 8682 | . . . . 5 ⊢ ((2o ↑m ℕ) ↑m ℕ) ≈ (2o ↑m ℕ) |
26 | 25, 11 | entr4i 8685 | . . . 4 ⊢ ((2o ↑m ℕ) ↑m ℕ) ≈ 𝒫 ℕ |
27 | domentr 8687 | . . . 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 692 | . . 3 ⊢ (ℚ ↑m ℕ) ≼ 𝒫 ℕ |
29 | domtr 8681 | . . 3 ⊢ ((ℝ ≼ (ℚ ↑m ℕ) ∧ (ℚ ↑m ℕ) ≼ 𝒫 ℕ) → ℝ ≼ 𝒫 ℕ) | |
30 | 3, 28, 29 | mp2an 692 | . 2 ⊢ ℝ ≼ 𝒫 ℕ |
31 | rpnnen2 15787 | . . 3 ⊢ 𝒫 ℕ ≼ (0[,]1) | |
32 | reex 10820 | . . . 4 ⊢ ℝ ∈ V | |
33 | unitssre 13087 | . . . 4 ⊢ (0[,]1) ⊆ ℝ | |
34 | ssdomg 8674 | . . . 4 ⊢ (ℝ ∈ V → ((0[,]1) ⊆ ℝ → (0[,]1) ≼ ℝ)) | |
35 | 32, 33, 34 | mp2 9 | . . 3 ⊢ (0[,]1) ≼ ℝ |
36 | domtr 8681 | . . 3 ⊢ ((𝒫 ℕ ≼ (0[,]1) ∧ (0[,]1) ≼ ℝ) → 𝒫 ℕ ≼ ℝ) | |
37 | 31, 35, 36 | mp2an 692 | . 2 ⊢ 𝒫 ℕ ≼ ℝ |
38 | sbth 8766 | . 2 ⊢ ((ℝ ≼ 𝒫 ℕ ∧ 𝒫 ℕ ≼ ℝ) → ℝ ≈ 𝒫 ℕ) | |
39 | 30, 37, 38 | mp2an 692 | 1 ⊢ ℝ ≈ 𝒫 ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 Vcvv 3408 ⊆ wss 3866 𝒫 cpw 4513 class class class wbr 5053 × cxp 5549 (class class class)co 7213 ωcom 7644 2oc2o 8196 ↑m cmap 8508 ≈ cen 8623 ≼ cdom 8624 ≺ csdm 8625 ℝcr 10728 0cc0 10729 1c1 10730 ℕcn 11830 ℚcq 12544 [,]cicc 12938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-oadd 8206 df-omul 8207 df-er 8391 df-map 8510 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-acn 9558 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-q 12545 df-rp 12587 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-limsup 15032 df-clim 15049 df-rlim 15050 df-sum 15250 |
This theorem is referenced by: rexpen 15789 cpnnen 15790 rucALT 15791 cnso 15808 2ndcredom 22347 opnreen 23728 |
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