![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 30675 | . . . . . 6 ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂) |
4 | 3 | sselda 3744 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (toCaraSiga‘𝑀)) → 𝑎 ∈ 𝒫 𝑂) |
5 | 4 | elpwid 4314 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (toCaraSiga‘𝑀)) → 𝑎 ⊆ 𝑂) |
6 | 5 | ralrimiva 3104 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ (toCaraSiga‘𝑀)𝑎 ⊆ 𝑂) |
7 | unissb 4621 | . . 3 ⊢ (∪ (toCaraSiga‘𝑀) ⊆ 𝑂 ↔ ∀𝑎 ∈ (toCaraSiga‘𝑀)𝑎 ⊆ 𝑂) | |
8 | 6, 7 | sylibr 224 | . 2 ⊢ (𝜑 → ∪ (toCaraSiga‘𝑀) ⊆ 𝑂) |
9 | baselcarsg.1 | . . 3 ⊢ (𝜑 → (𝑀‘∅) = 0) | |
10 | 1, 2, 9 | baselcarsg 30677 | . 2 ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀)) |
11 | unissel 4620 | . 2 ⊢ ((∪ (toCaraSiga‘𝑀) ⊆ 𝑂 ∧ 𝑂 ∈ (toCaraSiga‘𝑀)) → ∪ (toCaraSiga‘𝑀) = 𝑂) | |
12 | 8, 10, 11 | syl2anc 696 | 1 ⊢ (𝜑 → ∪ (toCaraSiga‘𝑀) = 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∀wral 3050 ⊆ wss 3715 ∅c0 4058 𝒫 cpw 4302 ∪ cuni 4588 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 0cc0 10128 +∞cpnf 10263 [,]cicc 12371 toCaraSigaccarsg 30672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-1st 7333 df-2nd 7334 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-xadd 12140 df-icc 12375 df-carsg 30673 |
This theorem is referenced by: carsgclctun 30692 |
Copyright terms: Public domain | W3C validator |