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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 31198 | . . . . . 6 ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂) |
4 | 3 | sselda 3857 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (toCaraSiga‘𝑀)) → 𝑎 ∈ 𝒫 𝑂) |
5 | 4 | elpwid 4432 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (toCaraSiga‘𝑀)) → 𝑎 ⊆ 𝑂) |
6 | 5 | ralrimiva 3129 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ (toCaraSiga‘𝑀)𝑎 ⊆ 𝑂) |
7 | unissb 4741 | . . 3 ⊢ (∪ (toCaraSiga‘𝑀) ⊆ 𝑂 ↔ ∀𝑎 ∈ (toCaraSiga‘𝑀)𝑎 ⊆ 𝑂) | |
8 | 6, 7 | sylibr 226 | . 2 ⊢ (𝜑 → ∪ (toCaraSiga‘𝑀) ⊆ 𝑂) |
9 | baselcarsg.1 | . . 3 ⊢ (𝜑 → (𝑀‘∅) = 0) | |
10 | 1, 2, 9 | baselcarsg 31200 | . 2 ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀)) |
11 | unissel 4740 | . 2 ⊢ ((∪ (toCaraSiga‘𝑀) ⊆ 𝑂 ∧ 𝑂 ∈ (toCaraSiga‘𝑀)) → ∪ (toCaraSiga‘𝑀) = 𝑂) | |
12 | 8, 10, 11 | syl2anc 576 | 1 ⊢ (𝜑 → ∪ (toCaraSiga‘𝑀) = 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ∀wral 3085 ⊆ wss 3828 ∅c0 4177 𝒫 cpw 4420 ∪ cuni 4710 ⟶wf 6182 ‘cfv 6186 (class class class)co 6974 0cc0 10331 +∞cpnf 10467 [,]cicc 12554 toCaraSigaccarsg 31195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2747 ax-rep 5047 ax-sep 5058 ax-nul 5065 ax-pow 5117 ax-pr 5184 ax-un 7277 ax-cnex 10387 ax-resscn 10388 ax-1cn 10389 ax-icn 10390 ax-addcl 10391 ax-addrcl 10392 ax-mulcl 10393 ax-mulrcl 10394 ax-mulcom 10395 ax-addass 10396 ax-mulass 10397 ax-distr 10398 ax-i2m1 10399 ax-1ne0 10400 ax-1rid 10401 ax-rnegex 10402 ax-rrecex 10403 ax-cnre 10404 ax-pre-lttri 10405 ax-pre-lttrn 10406 ax-pre-ltadd 10407 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2756 df-cleq 2768 df-clel 2843 df-nfc 2915 df-ne 2965 df-nel 3071 df-ral 3090 df-rex 3091 df-reu 3092 df-rab 3094 df-v 3414 df-sbc 3681 df-csb 3786 df-dif 3831 df-un 3833 df-in 3835 df-ss 3842 df-nul 4178 df-if 4349 df-pw 4422 df-sn 4440 df-pr 4442 df-op 4446 df-uni 4711 df-iun 4792 df-br 4928 df-opab 4990 df-mpt 5007 df-id 5309 df-po 5323 df-so 5324 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-ov 6977 df-oprab 6978 df-mpo 6979 df-1st 7498 df-2nd 7499 df-er 8085 df-en 8303 df-dom 8304 df-sdom 8305 df-pnf 10472 df-mnf 10473 df-xr 10474 df-ltxr 10475 df-xadd 12322 df-icc 12558 df-carsg 31196 |
This theorem is referenced by: carsgclctun 31215 |
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