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 32255 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑂) |
5 | difin2 4230 | . . 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 32256 | . . 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 32257 | . 2 ⊢ (𝜑 → ((𝑂 ∖ 𝐵) ∩ 𝐴) ∈ (toCaraSiga‘𝑀)) |
11 | 6, 10 | eqeltrd 2840 | 1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ (toCaraSiga‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ∖ cdif 3888 ∪ cun 3889 ∩ cin 3890 ⊆ wss 3891 𝒫 cpw 4538 class class class wbr 5078 ⟶wf 6426 ‘cfv 6430 (class class class)co 7268 0cc0 10855 +∞cpnf 10990 ≤ cle 10994 +𝑒 cxad 12828 [,]cicc 13064 toCaraSigaccarsg 32247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-po 5502 df-so 5503 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-ov 7271 df-oprab 7272 df-mpo 7273 df-1st 7817 df-2nd 7818 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-xadd 12831 df-icc 13068 df-carsg 32248 |
This theorem is referenced by: carsgclctunlem3 32266 |
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