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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 16186, which only asserts that ℝ is uncountable, i.e. has a cardinality larger than ω. The main proof is in two parts, rpnnen1 12967 and rpnnen2 16169, each showing an injection in one direction, and this last part uses sbth 9093 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 12218 | . . . 4 ⊢ ℕ ∈ V | |
2 | qex 12945 | . . . 4 ⊢ ℚ ∈ V | |
3 | 1, 2 | rpnnen1 12967 | . . 3 ⊢ ℝ ≼ (ℚ ↑m ℕ) |
4 | qnnen 16156 | . . . . . . 7 ⊢ ℚ ≈ ℕ | |
5 | 1 | canth2 9130 | . . . . . . 7 ⊢ ℕ ≺ 𝒫 ℕ |
6 | ensdomtr 9113 | . . . . . . 7 ⊢ ((ℚ ≈ ℕ ∧ ℕ ≺ 𝒫 ℕ) → ℚ ≺ 𝒫 ℕ) | |
7 | 4, 5, 6 | mp2an 691 | . . . . . 6 ⊢ ℚ ≺ 𝒫 ℕ |
8 | sdomdom 8976 | . . . . . 6 ⊢ (ℚ ≺ 𝒫 ℕ → ℚ ≼ 𝒫 ℕ) | |
9 | mapdom1 9142 | . . . . . 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 9079 | . . . . . 6 ⊢ 𝒫 ℕ ≈ (2o ↑m ℕ) |
12 | 1 | enref 8981 | . . . . . 6 ⊢ ℕ ≈ ℕ |
13 | mapen 9141 | . . . . . 6 ⊢ ((𝒫 ℕ ≈ (2o ↑m ℕ) ∧ ℕ ≈ ℕ) → (𝒫 ℕ ↑m ℕ) ≈ ((2o ↑m ℕ) ↑m ℕ)) | |
14 | 11, 12, 13 | mp2an 691 | . . . . 5 ⊢ (𝒫 ℕ ↑m ℕ) ≈ ((2o ↑m ℕ) ↑m ℕ) |
15 | domentr 9009 | . . . . 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 691 | . . . 4 ⊢ (ℚ ↑m ℕ) ≼ ((2o ↑m ℕ) ↑m ℕ) |
17 | 2onn 8641 | . . . . . . 7 ⊢ 2o ∈ ω | |
18 | mapxpen 9143 | . . . . . . 7 ⊢ ((2o ∈ ω ∧ ℕ ∈ V ∧ ℕ ∈ V) → ((2o ↑m ℕ) ↑m ℕ) ≈ (2o ↑m (ℕ × ℕ))) | |
19 | 17, 1, 1, 18 | mp3an 1462 | . . . . . 6 ⊢ ((2o ↑m ℕ) ↑m ℕ) ≈ (2o ↑m (ℕ × ℕ)) |
20 | 17 | elexi 3494 | . . . . . . . 8 ⊢ 2o ∈ V |
21 | 20 | enref 8981 | . . . . . . 7 ⊢ 2o ≈ 2o |
22 | xpnnen 16154 | . . . . . . 7 ⊢ (ℕ × ℕ) ≈ ℕ | |
23 | mapen 9141 | . . . . . . 7 ⊢ ((2o ≈ 2o ∧ (ℕ × ℕ) ≈ ℕ) → (2o ↑m (ℕ × ℕ)) ≈ (2o ↑m ℕ)) | |
24 | 21, 22, 23 | mp2an 691 | . . . . . 6 ⊢ (2o ↑m (ℕ × ℕ)) ≈ (2o ↑m ℕ) |
25 | 19, 24 | entri 9004 | . . . . 5 ⊢ ((2o ↑m ℕ) ↑m ℕ) ≈ (2o ↑m ℕ) |
26 | 25, 11 | entr4i 9007 | . . . 4 ⊢ ((2o ↑m ℕ) ↑m ℕ) ≈ 𝒫 ℕ |
27 | domentr 9009 | . . . 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 691 | . . 3 ⊢ (ℚ ↑m ℕ) ≼ 𝒫 ℕ |
29 | domtr 9003 | . . 3 ⊢ ((ℝ ≼ (ℚ ↑m ℕ) ∧ (ℚ ↑m ℕ) ≼ 𝒫 ℕ) → ℝ ≼ 𝒫 ℕ) | |
30 | 3, 28, 29 | mp2an 691 | . 2 ⊢ ℝ ≼ 𝒫 ℕ |
31 | rpnnen2 16169 | . . 3 ⊢ 𝒫 ℕ ≼ (0[,]1) | |
32 | reex 11201 | . . . 4 ⊢ ℝ ∈ V | |
33 | unitssre 13476 | . . . 4 ⊢ (0[,]1) ⊆ ℝ | |
34 | ssdomg 8996 | . . . 4 ⊢ (ℝ ∈ V → ((0[,]1) ⊆ ℝ → (0[,]1) ≼ ℝ)) | |
35 | 32, 33, 34 | mp2 9 | . . 3 ⊢ (0[,]1) ≼ ℝ |
36 | domtr 9003 | . . 3 ⊢ ((𝒫 ℕ ≼ (0[,]1) ∧ (0[,]1) ≼ ℝ) → 𝒫 ℕ ≼ ℝ) | |
37 | 31, 35, 36 | mp2an 691 | . 2 ⊢ 𝒫 ℕ ≼ ℝ |
38 | sbth 9093 | . 2 ⊢ ((ℝ ≼ 𝒫 ℕ ∧ 𝒫 ℕ ≼ ℝ) → ℝ ≈ 𝒫 ℕ) | |
39 | 30, 37, 38 | mp2an 691 | 1 ⊢ ℝ ≈ 𝒫 ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3475 ⊆ wss 3949 𝒫 cpw 4603 class class class wbr 5149 × cxp 5675 (class class class)co 7409 ωcom 7855 2oc2o 8460 ↑m cmap 8820 ≈ cen 8936 ≼ cdom 8937 ≺ csdm 8938 ℝcr 11109 0cc0 11110 1c1 11111 ℕcn 12212 ℚcq 12932 [,]cicc 13327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 |
This theorem is referenced by: rexpen 16171 cpnnen 16172 rucALT 16173 cnso 16190 2ndcredom 22954 opnreen 24347 |
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