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Mirrors > Home > MPE Home > Th. List > ovolctb2 | Structured version Visualization version GIF version |
Description: The volume of a countable set is 0. (Contributed by Mario Carneiro, 17-Mar-2014.) |
Ref | Expression |
---|---|
ovolctb2 | ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (vol*‘𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 4147 | . 2 ⊢ 𝐴 ⊆ (𝐴 ∪ ℕ) | |
2 | simpl 485 | . . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → 𝐴 ⊆ ℝ) | |
3 | nnssre 11636 | . . 3 ⊢ ℕ ⊆ ℝ | |
4 | unss 4159 | . . 3 ⊢ ((𝐴 ⊆ ℝ ∧ ℕ ⊆ ℝ) ↔ (𝐴 ∪ ℕ) ⊆ ℝ) | |
5 | 2, 3, 4 | sylanblc 591 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (𝐴 ∪ ℕ) ⊆ ℝ) |
6 | nnenom 13342 | . . . . . . . 8 ⊢ ℕ ≈ ω | |
7 | domentr 8562 | . . . . . . . 8 ⊢ ((𝐴 ≼ ℕ ∧ ℕ ≈ ω) → 𝐴 ≼ ω) | |
8 | 6, 7 | mpan2 689 | . . . . . . 7 ⊢ (𝐴 ≼ ℕ → 𝐴 ≼ ω) |
9 | 8 | adantl 484 | . . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → 𝐴 ≼ ω) |
10 | nnct 13343 | . . . . . 6 ⊢ ℕ ≼ ω | |
11 | unctb 9621 | . . . . . 6 ⊢ ((𝐴 ≼ ω ∧ ℕ ≼ ω) → (𝐴 ∪ ℕ) ≼ ω) | |
12 | 9, 10, 11 | sylancl 588 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (𝐴 ∪ ℕ) ≼ ω) |
13 | 6 | ensymi 8553 | . . . . 5 ⊢ ω ≈ ℕ |
14 | domentr 8562 | . . . . 5 ⊢ (((𝐴 ∪ ℕ) ≼ ω ∧ ω ≈ ℕ) → (𝐴 ∪ ℕ) ≼ ℕ) | |
15 | 12, 13, 14 | sylancl 588 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (𝐴 ∪ ℕ) ≼ ℕ) |
16 | reex 10622 | . . . . . . 7 ⊢ ℝ ∈ V | |
17 | 16 | ssex 5217 | . . . . . 6 ⊢ ((𝐴 ∪ ℕ) ⊆ ℝ → (𝐴 ∪ ℕ) ∈ V) |
18 | 5, 17 | syl 17 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (𝐴 ∪ ℕ) ∈ V) |
19 | ssun2 4148 | . . . . 5 ⊢ ℕ ⊆ (𝐴 ∪ ℕ) | |
20 | ssdomg 8549 | . . . . 5 ⊢ ((𝐴 ∪ ℕ) ∈ V → (ℕ ⊆ (𝐴 ∪ ℕ) → ℕ ≼ (𝐴 ∪ ℕ))) | |
21 | 18, 19, 20 | mpisyl 21 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → ℕ ≼ (𝐴 ∪ ℕ)) |
22 | sbth 8631 | . . . 4 ⊢ (((𝐴 ∪ ℕ) ≼ ℕ ∧ ℕ ≼ (𝐴 ∪ ℕ)) → (𝐴 ∪ ℕ) ≈ ℕ) | |
23 | 15, 21, 22 | syl2anc 586 | . . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (𝐴 ∪ ℕ) ≈ ℕ) |
24 | ovolctb 24085 | . . 3 ⊢ (((𝐴 ∪ ℕ) ⊆ ℝ ∧ (𝐴 ∪ ℕ) ≈ ℕ) → (vol*‘(𝐴 ∪ ℕ)) = 0) | |
25 | 5, 23, 24 | syl2anc 586 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (vol*‘(𝐴 ∪ ℕ)) = 0) |
26 | ovolssnul 24082 | . 2 ⊢ ((𝐴 ⊆ (𝐴 ∪ ℕ) ∧ (𝐴 ∪ ℕ) ⊆ ℝ ∧ (vol*‘(𝐴 ∪ ℕ)) = 0) → (vol*‘𝐴) = 0) | |
27 | 1, 5, 25, 26 | mp3an2i 1462 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (vol*‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∪ cun 3933 ⊆ wss 3935 class class class wbr 5058 ‘cfv 6349 ωcom 7574 ≈ cen 8500 ≼ cdom 8501 ℝcr 10530 0cc0 10531 ℕcn 11632 vol*covol 24057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xadd 12502 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-xmet 20532 df-met 20533 df-ovol 24059 |
This theorem is referenced by: ovol0 24088 ovolfi 24089 uniiccdif 24173 voliunnfl 34930 volsupnfl 34931 |
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