![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 16192, which only asserts that ℝ is uncountable, i.e. has a cardinality larger than ω. The main proof is in two parts, rpnnen1 12973 and rpnnen2 16175, each showing an injection in one direction, and this last part uses sbth 9097 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 12224 | . . . 4 ⊢ ℕ ∈ V | |
2 | qex 12951 | . . . 4 ⊢ ℚ ∈ V | |
3 | 1, 2 | rpnnen1 12973 | . . 3 ⊢ ℝ ≼ (ℚ ↑m ℕ) |
4 | qnnen 16162 | . . . . . . 7 ⊢ ℚ ≈ ℕ | |
5 | 1 | canth2 9134 | . . . . . . 7 ⊢ ℕ ≺ 𝒫 ℕ |
6 | ensdomtr 9117 | . . . . . . 7 ⊢ ((ℚ ≈ ℕ ∧ ℕ ≺ 𝒫 ℕ) → ℚ ≺ 𝒫 ℕ) | |
7 | 4, 5, 6 | mp2an 688 | . . . . . 6 ⊢ ℚ ≺ 𝒫 ℕ |
8 | sdomdom 8980 | . . . . . 6 ⊢ (ℚ ≺ 𝒫 ℕ → ℚ ≼ 𝒫 ℕ) | |
9 | mapdom1 9146 | . . . . . 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 9083 | . . . . . 6 ⊢ 𝒫 ℕ ≈ (2o ↑m ℕ) |
12 | 1 | enref 8985 | . . . . . 6 ⊢ ℕ ≈ ℕ |
13 | mapen 9145 | . . . . . 6 ⊢ ((𝒫 ℕ ≈ (2o ↑m ℕ) ∧ ℕ ≈ ℕ) → (𝒫 ℕ ↑m ℕ) ≈ ((2o ↑m ℕ) ↑m ℕ)) | |
14 | 11, 12, 13 | mp2an 688 | . . . . 5 ⊢ (𝒫 ℕ ↑m ℕ) ≈ ((2o ↑m ℕ) ↑m ℕ) |
15 | domentr 9013 | . . . . 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 688 | . . . 4 ⊢ (ℚ ↑m ℕ) ≼ ((2o ↑m ℕ) ↑m ℕ) |
17 | 2onn 8645 | . . . . . . 7 ⊢ 2o ∈ ω | |
18 | mapxpen 9147 | . . . . . . 7 ⊢ ((2o ∈ ω ∧ ℕ ∈ V ∧ ℕ ∈ V) → ((2o ↑m ℕ) ↑m ℕ) ≈ (2o ↑m (ℕ × ℕ))) | |
19 | 17, 1, 1, 18 | mp3an 1459 | . . . . . 6 ⊢ ((2o ↑m ℕ) ↑m ℕ) ≈ (2o ↑m (ℕ × ℕ)) |
20 | 17 | elexi 3492 | . . . . . . . 8 ⊢ 2o ∈ V |
21 | 20 | enref 8985 | . . . . . . 7 ⊢ 2o ≈ 2o |
22 | xpnnen 16160 | . . . . . . 7 ⊢ (ℕ × ℕ) ≈ ℕ | |
23 | mapen 9145 | . . . . . . 7 ⊢ ((2o ≈ 2o ∧ (ℕ × ℕ) ≈ ℕ) → (2o ↑m (ℕ × ℕ)) ≈ (2o ↑m ℕ)) | |
24 | 21, 22, 23 | mp2an 688 | . . . . . 6 ⊢ (2o ↑m (ℕ × ℕ)) ≈ (2o ↑m ℕ) |
25 | 19, 24 | entri 9008 | . . . . 5 ⊢ ((2o ↑m ℕ) ↑m ℕ) ≈ (2o ↑m ℕ) |
26 | 25, 11 | entr4i 9011 | . . . 4 ⊢ ((2o ↑m ℕ) ↑m ℕ) ≈ 𝒫 ℕ |
27 | domentr 9013 | . . . 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 688 | . . 3 ⊢ (ℚ ↑m ℕ) ≼ 𝒫 ℕ |
29 | domtr 9007 | . . 3 ⊢ ((ℝ ≼ (ℚ ↑m ℕ) ∧ (ℚ ↑m ℕ) ≼ 𝒫 ℕ) → ℝ ≼ 𝒫 ℕ) | |
30 | 3, 28, 29 | mp2an 688 | . 2 ⊢ ℝ ≼ 𝒫 ℕ |
31 | rpnnen2 16175 | . . 3 ⊢ 𝒫 ℕ ≼ (0[,]1) | |
32 | reex 11205 | . . . 4 ⊢ ℝ ∈ V | |
33 | unitssre 13482 | . . . 4 ⊢ (0[,]1) ⊆ ℝ | |
34 | ssdomg 9000 | . . . 4 ⊢ (ℝ ∈ V → ((0[,]1) ⊆ ℝ → (0[,]1) ≼ ℝ)) | |
35 | 32, 33, 34 | mp2 9 | . . 3 ⊢ (0[,]1) ≼ ℝ |
36 | domtr 9007 | . . 3 ⊢ ((𝒫 ℕ ≼ (0[,]1) ∧ (0[,]1) ≼ ℝ) → 𝒫 ℕ ≼ ℝ) | |
37 | 31, 35, 36 | mp2an 688 | . 2 ⊢ 𝒫 ℕ ≼ ℝ |
38 | sbth 9097 | . 2 ⊢ ((ℝ ≼ 𝒫 ℕ ∧ 𝒫 ℕ ≼ ℝ) → ℝ ≈ 𝒫 ℕ) | |
39 | 30, 37, 38 | mp2an 688 | 1 ⊢ ℝ ≈ 𝒫 ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2104 Vcvv 3472 ⊆ wss 3949 𝒫 cpw 4603 class class class wbr 5149 × cxp 5675 (class class class)co 7413 ωcom 7859 2oc2o 8464 ↑m cmap 8824 ≈ cen 8940 ≼ cdom 8941 ≺ csdm 8942 ℝcr 11113 0cc0 11114 1c1 11115 ℕcn 12218 ℚcq 12938 [,]cicc 13333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-omul 8475 df-er 8707 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-inf 9442 df-oi 9509 df-card 9938 df-acn 9941 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-div 11878 df-nn 12219 df-2 12281 df-3 12282 df-n0 12479 df-z 12565 df-uz 12829 df-q 12939 df-rp 12981 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-seq 13973 df-exp 14034 df-hash 14297 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 |
This theorem is referenced by: rexpen 16177 cpnnen 16178 rucALT 16179 cnso 16196 2ndcredom 23176 opnreen 24569 |
Copyright terms: Public domain | W3C validator |