![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > elcarsgss | Structured version Visualization version GIF version |
Description: Caratheodory measurable sets are subsets of the universe. (Contributed by Thierry Arnoux, 21-May-2020.) |
Ref | Expression |
---|---|
carsgval.1 | ⊢ (𝜑 → 𝑂 ∈ 𝑉) |
carsgval.2 | ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) |
difelcarsg.1 | ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) |
Ref | Expression |
---|---|
elcarsgss | ⊢ (𝜑 → 𝐴 ⊆ 𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carsgval.1 | . . . 4 ⊢ (𝜑 → 𝑂 ∈ 𝑉) | |
2 | carsgval.2 | . . . 4 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) | |
3 | 1, 2 | carsgcl 33240 | . . 3 ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂) |
4 | difelcarsg.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) | |
5 | 3, 4 | sseldd 3981 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑂) |
6 | 5 | elpwid 4609 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ⊆ wss 3946 𝒫 cpw 4600 ⟶wf 6535 ‘cfv 6539 (class class class)co 7403 0cc0 11105 +∞cpnf 11240 [,]cicc 13322 toCaraSigaccarsg 33237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-ov 7406 df-carsg 33238 |
This theorem is referenced by: unelcarsg 33248 difelcarsg2 33249 |
Copyright terms: Public domain | W3C validator |