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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > carsguni | Structured version Visualization version GIF version |
Description: The union of all Caratheodory measurable sets is the universe. (Contributed by Thierry Arnoux, 22-May-2020.) |
Ref | Expression |
---|---|
carsgval.1 | ⊢ (𝜑 → 𝑂 ∈ 𝑉) |
carsgval.2 | ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) |
baselcarsg.1 | ⊢ (𝜑 → (𝑀‘∅) = 0) |
Ref | Expression |
---|---|
carsguni | ⊢ (𝜑 → ∪ (toCaraSiga‘𝑀) = 𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carsgval.1 | . . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ 𝑉) | |
2 | carsgval.2 | . . . . . . 7 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) | |
3 | 1, 2 | carsgcl 33291 | . . . . . 6 ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂) |
4 | 3 | sselda 3981 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (toCaraSiga‘𝑀)) → 𝑎 ∈ 𝒫 𝑂) |
5 | 4 | elpwid 4610 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (toCaraSiga‘𝑀)) → 𝑎 ⊆ 𝑂) |
6 | 5 | ralrimiva 3146 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ (toCaraSiga‘𝑀)𝑎 ⊆ 𝑂) |
7 | unissb 4942 | . . 3 ⊢ (∪ (toCaraSiga‘𝑀) ⊆ 𝑂 ↔ ∀𝑎 ∈ (toCaraSiga‘𝑀)𝑎 ⊆ 𝑂) | |
8 | 6, 7 | sylibr 233 | . 2 ⊢ (𝜑 → ∪ (toCaraSiga‘𝑀) ⊆ 𝑂) |
9 | baselcarsg.1 | . . 3 ⊢ (𝜑 → (𝑀‘∅) = 0) | |
10 | 1, 2, 9 | baselcarsg 33293 | . 2 ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀)) |
11 | unissel 4941 | . 2 ⊢ ((∪ (toCaraSiga‘𝑀) ⊆ 𝑂 ∧ 𝑂 ∈ (toCaraSiga‘𝑀)) → ∪ (toCaraSiga‘𝑀) = 𝑂) | |
12 | 8, 10, 11 | syl2anc 584 | 1 ⊢ (𝜑 → ∪ (toCaraSiga‘𝑀) = 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 ∪ cuni 4907 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 0cc0 11106 +∞cpnf 11241 [,]cicc 13323 toCaraSigaccarsg 33288 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-xadd 13089 df-icc 13327 df-carsg 33289 |
This theorem is referenced by: carsgclctun 33308 |
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