![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > saliuncl | Structured version Visualization version GIF version |
Description: SAlg sigma-algebra is closed under countable indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
saliuncl.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
saliuncl.kct | ⊢ (𝜑 → 𝐾 ≼ ω) |
saliuncl.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ∈ 𝑆) |
Ref | Expression |
---|---|
saliuncl | ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1916 | . 2 ⊢ Ⅎ𝑘𝜑 | |
2 | nfcv 2902 | . 2 ⊢ Ⅎ𝑘𝑆 | |
3 | nfcv 2902 | . 2 ⊢ Ⅎ𝑘𝐾 | |
4 | saliuncl.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
5 | saliuncl.kct | . 2 ⊢ (𝜑 → 𝐾 ≼ ω) | |
6 | saliuncl.b | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ∈ 𝑆) | |
7 | 1, 2, 3, 4, 5, 6 | saliunclf 45497 | 1 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ∪ ciun 4997 class class class wbr 5148 ωcom 7859 ≼ cdom 8943 SAlgcsalg 45483 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-card 9940 df-acn 9943 df-salg 45484 |
This theorem is referenced by: subsaliuncl 45533 meaiunlelem 45643 meaiuninclem 45655 meaiuninc3v 45659 meaiininclem 45661 caratheodory 45703 opnvonmbllem2 45808 ctvonmbl 45864 vonct 45868 smfaddlem2 45939 smflimlem1 45946 smfresal 45963 smfmullem4 45969 |
Copyright terms: Public domain | W3C validator |