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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 4062 | . 2 ⊢ 𝐴 ⊆ (𝐴 ∪ ℕ) | |
2 | simpl 486 | . . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → 𝐴 ⊆ ℝ) | |
3 | nnssre 11720 | . . 3 ⊢ ℕ ⊆ ℝ | |
4 | unss 4074 | . . 3 ⊢ ((𝐴 ⊆ ℝ ∧ ℕ ⊆ ℝ) ↔ (𝐴 ∪ ℕ) ⊆ ℝ) | |
5 | 2, 3, 4 | sylanblc 592 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (𝐴 ∪ ℕ) ⊆ ℝ) |
6 | nnenom 13439 | . . . . . . . 8 ⊢ ℕ ≈ ω | |
7 | domentr 8614 | . . . . . . . 8 ⊢ ((𝐴 ≼ ℕ ∧ ℕ ≈ ω) → 𝐴 ≼ ω) | |
8 | 6, 7 | mpan2 691 | . . . . . . 7 ⊢ (𝐴 ≼ ℕ → 𝐴 ≼ ω) |
9 | 8 | adantl 485 | . . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → 𝐴 ≼ ω) |
10 | nnct 13440 | . . . . . 6 ⊢ ℕ ≼ ω | |
11 | unctb 9705 | . . . . . 6 ⊢ ((𝐴 ≼ ω ∧ ℕ ≼ ω) → (𝐴 ∪ ℕ) ≼ ω) | |
12 | 9, 10, 11 | sylancl 589 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (𝐴 ∪ ℕ) ≼ ω) |
13 | 6 | ensymi 8605 | . . . . 5 ⊢ ω ≈ ℕ |
14 | domentr 8614 | . . . . 5 ⊢ (((𝐴 ∪ ℕ) ≼ ω ∧ ω ≈ ℕ) → (𝐴 ∪ ℕ) ≼ ℕ) | |
15 | 12, 13, 14 | sylancl 589 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (𝐴 ∪ ℕ) ≼ ℕ) |
16 | reex 10706 | . . . . . . 7 ⊢ ℝ ∈ V | |
17 | 16 | ssex 5189 | . . . . . 6 ⊢ ((𝐴 ∪ ℕ) ⊆ ℝ → (𝐴 ∪ ℕ) ∈ V) |
18 | 5, 17 | syl 17 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (𝐴 ∪ ℕ) ∈ V) |
19 | ssun2 4063 | . . . . 5 ⊢ ℕ ⊆ (𝐴 ∪ ℕ) | |
20 | ssdomg 8601 | . . . . 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 8687 | . . . 4 ⊢ (((𝐴 ∪ ℕ) ≼ ℕ ∧ ℕ ≼ (𝐴 ∪ ℕ)) → (𝐴 ∪ ℕ) ≈ ℕ) | |
23 | 15, 21, 22 | syl2anc 587 | . . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (𝐴 ∪ ℕ) ≈ ℕ) |
24 | ovolctb 24242 | . . 3 ⊢ (((𝐴 ∪ ℕ) ⊆ ℝ ∧ (𝐴 ∪ ℕ) ≈ ℕ) → (vol*‘(𝐴 ∪ ℕ)) = 0) | |
25 | 5, 23, 24 | syl2anc 587 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (vol*‘(𝐴 ∪ ℕ)) = 0) |
26 | ovolssnul 24239 | . 2 ⊢ ((𝐴 ⊆ (𝐴 ∪ ℕ) ∧ (𝐴 ∪ ℕ) ⊆ ℝ ∧ (vol*‘(𝐴 ∪ ℕ)) = 0) → (vol*‘𝐴) = 0) | |
27 | 1, 5, 25, 26 | mp3an2i 1467 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → (vol*‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ∪ cun 3841 ⊆ wss 3843 class class class wbr 5030 ‘cfv 6339 ωcom 7599 ≈ cen 8552 ≼ cdom 8553 ℝcr 10614 0cc0 10615 ℕcn 11716 vol*covol 24214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-inf 8980 df-oi 9047 df-dju 9403 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-z 12063 df-uz 12325 df-q 12431 df-rp 12473 df-xadd 12591 df-ioo 12825 df-ico 12827 df-icc 12828 df-fz 12982 df-fzo 13125 df-seq 13461 df-exp 13522 df-hash 13783 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-clim 14935 df-sum 15136 df-xmet 20210 df-met 20211 df-ovol 24216 |
This theorem is referenced by: ovol0 24245 ovolfi 24246 uniiccdif 24330 voliunnfl 35444 volsupnfl 35445 |
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