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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > difelcarsg2 | Structured version Visualization version GIF version |
Description: The Caratheodory-measurable sets are closed under class difference. (Contributed by Thierry Arnoux, 30-May-2020.) |
Ref | Expression |
---|---|
carsgval.1 | ⊢ (𝜑 → 𝑂 ∈ 𝑉) |
carsgval.2 | ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) |
difelcarsg.1 | ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) |
inelcarsg.1 | ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏))) |
inelcarsg.2 | ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀)) |
Ref | Expression |
---|---|
difelcarsg2 | ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ (toCaraSiga‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carsgval.1 | . . . 4 ⊢ (𝜑 → 𝑂 ∈ 𝑉) | |
2 | carsgval.2 | . . . 4 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) | |
3 | difelcarsg.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) | |
4 | 1, 2, 3 | elcarsgss 30499 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑂) |
5 | difin2 3923 | . . 3 ⊢ (𝐴 ⊆ 𝑂 → (𝐴 ∖ 𝐵) = ((𝑂 ∖ 𝐵) ∩ 𝐴)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 ∖ 𝐵) = ((𝑂 ∖ 𝐵) ∩ 𝐴)) |
7 | inelcarsg.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀)) | |
8 | 1, 2, 7 | difelcarsg 30500 | . . 3 ⊢ (𝜑 → (𝑂 ∖ 𝐵) ∈ (toCaraSiga‘𝑀)) |
9 | inelcarsg.1 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏))) | |
10 | 1, 2, 8, 9, 3 | inelcarsg 30501 | . 2 ⊢ (𝜑 → ((𝑂 ∖ 𝐵) ∩ 𝐴) ∈ (toCaraSiga‘𝑀)) |
11 | 6, 10 | eqeltrd 2730 | 1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ (toCaraSiga‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ∖ cdif 3604 ∪ cun 3605 ∩ cin 3606 ⊆ wss 3607 𝒫 cpw 4191 class class class wbr 4685 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 0cc0 9974 +∞cpnf 10109 ≤ cle 10113 +𝑒 cxad 11982 [,]cicc 12216 toCaraSigaccarsg 30491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-xadd 11985 df-icc 12220 df-carsg 30492 |
This theorem is referenced by: carsgclctunlem3 30510 |
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