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Mirrors > Home > MPE Home > Th. List > rectbntr0 | Structured version Visualization version GIF version |
Description: A countable subset of the reals has empty interior. (Contributed by Mario Carneiro, 26-Jul-2014.) |
Ref | Expression |
---|---|
rectbntr0 | ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → ((int‘(topGen‘ran (,)))‘𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnex 11631 | . . . 4 ⊢ ℕ ∈ V | |
2 | 1 | canth2 8654 | . . 3 ⊢ ℕ ≺ 𝒫 ℕ |
3 | domnsym 8627 | . . 3 ⊢ (𝒫 ℕ ≼ ℕ → ¬ ℕ ≺ 𝒫 ℕ) | |
4 | 2, 3 | mt2 203 | . 2 ⊢ ¬ 𝒫 ℕ ≼ ℕ |
5 | retop 23367 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
6 | simpl 486 | . . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → 𝐴 ⊆ ℝ) | |
7 | uniretop 23368 | . . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
8 | 7 | ntropn 21654 | . . . . . 6 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝐴 ⊆ ℝ) → ((int‘(topGen‘ran (,)))‘𝐴) ∈ (topGen‘ran (,))) |
9 | 5, 6, 8 | sylancr 590 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → ((int‘(topGen‘ran (,)))‘𝐴) ∈ (topGen‘ran (,))) |
10 | opnreen 23436 | . . . . . 6 ⊢ ((((int‘(topGen‘ran (,)))‘𝐴) ∈ (topGen‘ran (,)) ∧ ((int‘(topGen‘ran (,)))‘𝐴) ≠ ∅) → ((int‘(topGen‘ran (,)))‘𝐴) ≈ 𝒫 ℕ) | |
11 | 10 | ex 416 | . . . . 5 ⊢ (((int‘(topGen‘ran (,)))‘𝐴) ∈ (topGen‘ran (,)) → (((int‘(topGen‘ran (,)))‘𝐴) ≠ ∅ → ((int‘(topGen‘ran (,)))‘𝐴) ≈ 𝒫 ℕ)) |
12 | 9, 11 | syl 17 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (((int‘(topGen‘ran (,)))‘𝐴) ≠ ∅ → ((int‘(topGen‘ran (,)))‘𝐴) ≈ 𝒫 ℕ)) |
13 | reex 10617 | . . . . . . . 8 ⊢ ℝ ∈ V | |
14 | 13 | ssex 5189 | . . . . . . 7 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V) |
15 | 7 | ntrss2 21662 | . . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝐴 ⊆ ℝ) → ((int‘(topGen‘ran (,)))‘𝐴) ⊆ 𝐴) |
16 | 5, 15 | mpan 689 | . . . . . . 7 ⊢ (𝐴 ⊆ ℝ → ((int‘(topGen‘ran (,)))‘𝐴) ⊆ 𝐴) |
17 | ssdomg 8538 | . . . . . . 7 ⊢ (𝐴 ∈ V → (((int‘(topGen‘ran (,)))‘𝐴) ⊆ 𝐴 → ((int‘(topGen‘ran (,)))‘𝐴) ≼ 𝐴)) | |
18 | 14, 16, 17 | sylc 65 | . . . . . 6 ⊢ (𝐴 ⊆ ℝ → ((int‘(topGen‘ran (,)))‘𝐴) ≼ 𝐴) |
19 | domtr 8545 | . . . . . 6 ⊢ ((((int‘(topGen‘ran (,)))‘𝐴) ≼ 𝐴 ∧ 𝐴 ≼ ℕ) → ((int‘(topGen‘ran (,)))‘𝐴) ≼ ℕ) | |
20 | 18, 19 | sylan 583 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → ((int‘(topGen‘ran (,)))‘𝐴) ≼ ℕ) |
21 | ensym 8541 | . . . . 5 ⊢ (((int‘(topGen‘ran (,)))‘𝐴) ≈ 𝒫 ℕ → 𝒫 ℕ ≈ ((int‘(topGen‘ran (,)))‘𝐴)) | |
22 | endomtr 8550 | . . . . . 6 ⊢ ((𝒫 ℕ ≈ ((int‘(topGen‘ran (,)))‘𝐴) ∧ ((int‘(topGen‘ran (,)))‘𝐴) ≼ ℕ) → 𝒫 ℕ ≼ ℕ) | |
23 | 22 | expcom 417 | . . . . 5 ⊢ (((int‘(topGen‘ran (,)))‘𝐴) ≼ ℕ → (𝒫 ℕ ≈ ((int‘(topGen‘ran (,)))‘𝐴) → 𝒫 ℕ ≼ ℕ)) |
24 | 20, 21, 23 | syl2im 40 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (((int‘(topGen‘ran (,)))‘𝐴) ≈ 𝒫 ℕ → 𝒫 ℕ ≼ ℕ)) |
25 | 12, 24 | syld 47 | . . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (((int‘(topGen‘ran (,)))‘𝐴) ≠ ∅ → 𝒫 ℕ ≼ ℕ)) |
26 | 25 | necon1bd 3005 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (¬ 𝒫 ℕ ≼ ℕ → ((int‘(topGen‘ran (,)))‘𝐴) = ∅)) |
27 | 4, 26 | mpi 20 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → ((int‘(topGen‘ran (,)))‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 class class class wbr 5030 ran crn 5520 ‘cfv 6324 ≈ cen 8489 ≼ cdom 8490 ≺ csdm 8491 ℝcr 10525 ℕcn 11625 (,)cioo 12726 topGenctg 16703 Topctop 21498 intcnt 21622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-top 21499 df-topon 21516 df-bases 21551 df-ntr 21625 |
This theorem is referenced by: ioonct 42174 |
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