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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > saluncl | Structured version Visualization version GIF version |
Description: The union of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
saluncl | ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → (𝐸 ∪ 𝐹) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniprg 4818 | . . . 4 ⊢ ((𝐸 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → ∪ {𝐸, 𝐹} = (𝐸 ∪ 𝐹)) | |
2 | 1 | eqcomd 2804 | . . 3 ⊢ ((𝐸 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → (𝐸 ∪ 𝐹) = ∪ {𝐸, 𝐹}) |
3 | 2 | 3adant1 1127 | . 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → (𝐸 ∪ 𝐹) = ∪ {𝐸, 𝐹}) |
4 | prfi 8777 | . . . . 5 ⊢ {𝐸, 𝐹} ∈ Fin | |
5 | isfinite 9099 | . . . . . . 7 ⊢ ({𝐸, 𝐹} ∈ Fin ↔ {𝐸, 𝐹} ≺ ω) | |
6 | 5 | biimpi 219 | . . . . . 6 ⊢ ({𝐸, 𝐹} ∈ Fin → {𝐸, 𝐹} ≺ ω) |
7 | sdomdom 8520 | . . . . . 6 ⊢ ({𝐸, 𝐹} ≺ ω → {𝐸, 𝐹} ≼ ω) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ ({𝐸, 𝐹} ∈ Fin → {𝐸, 𝐹} ≼ ω) |
9 | 4, 8 | ax-mp 5 | . . . 4 ⊢ {𝐸, 𝐹} ≼ ω |
10 | 9 | a1i 11 | . . 3 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → {𝐸, 𝐹} ≼ ω) |
11 | prelpwi 5305 | . . . . 5 ⊢ ((𝐸 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → {𝐸, 𝐹} ∈ 𝒫 𝑆) | |
12 | 11 | 3adant1 1127 | . . . 4 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → {𝐸, 𝐹} ∈ 𝒫 𝑆) |
13 | issal 42956 | . . . . . . 7 ⊢ (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))) | |
14 | 13 | ibi 270 | . . . . . 6 ⊢ (𝑆 ∈ SAlg → (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))) |
15 | 14 | simp3d 1141 | . . . . 5 ⊢ (𝑆 ∈ SAlg → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)) |
16 | 15 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)) |
17 | breq1 5033 | . . . . . 6 ⊢ (𝑦 = {𝐸, 𝐹} → (𝑦 ≼ ω ↔ {𝐸, 𝐹} ≼ ω)) | |
18 | unieq 4811 | . . . . . . 7 ⊢ (𝑦 = {𝐸, 𝐹} → ∪ 𝑦 = ∪ {𝐸, 𝐹}) | |
19 | 18 | eleq1d 2874 | . . . . . 6 ⊢ (𝑦 = {𝐸, 𝐹} → (∪ 𝑦 ∈ 𝑆 ↔ ∪ {𝐸, 𝐹} ∈ 𝑆)) |
20 | 17, 19 | imbi12d 348 | . . . . 5 ⊢ (𝑦 = {𝐸, 𝐹} → ((𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆) ↔ ({𝐸, 𝐹} ≼ ω → ∪ {𝐸, 𝐹} ∈ 𝑆))) |
21 | 20 | rspcva 3569 | . . . 4 ⊢ (({𝐸, 𝐹} ∈ 𝒫 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)) → ({𝐸, 𝐹} ≼ ω → ∪ {𝐸, 𝐹} ∈ 𝑆)) |
22 | 12, 16, 21 | syl2anc 587 | . . 3 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → ({𝐸, 𝐹} ≼ ω → ∪ {𝐸, 𝐹} ∈ 𝑆)) |
23 | 10, 22 | mpd 15 | . 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → ∪ {𝐸, 𝐹} ∈ 𝑆) |
24 | 3, 23 | eqeltrd 2890 | 1 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → (𝐸 ∪ 𝐹) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∖ cdif 3878 ∪ cun 3879 ∅c0 4243 𝒫 cpw 4497 {cpr 4527 ∪ cuni 4800 class class class wbr 5030 ωcom 7560 ≼ cdom 8490 ≺ csdm 8491 Fincfn 8492 SAlgcsalg 42950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-salg 42951 |
This theorem is referenced by: salincl 42965 saluncld 42988 |
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