![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > unisalgen | Structured version Visualization version GIF version |
Description: The union of a set belongs to the sigma-algebra generated by the set. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
unisalgen.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
unisalgen.s | ⊢ 𝑆 = (SalGen‘𝑋) |
unisalgen.u | ⊢ 𝑈 = ∪ 𝑋 |
Ref | Expression |
---|---|
unisalgen | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unisalgen.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
2 | unisalgen.s | . . . 4 ⊢ 𝑆 = (SalGen‘𝑋) | |
3 | unisalgen.u | . . . 4 ⊢ 𝑈 = ∪ 𝑋 | |
4 | 1, 2, 3 | salgenuni 46293 | . . 3 ⊢ (𝜑 → ∪ 𝑆 = 𝑈) |
5 | 4 | eqcomd 2741 | . 2 ⊢ (𝜑 → 𝑈 = ∪ 𝑆) |
6 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑆 = (SalGen‘𝑋)) |
7 | salgencl 46288 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg) | |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (SalGen‘𝑋) ∈ SAlg) |
9 | 6, 8 | eqeltrd 2839 | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
10 | saluni 46281 | . . 3 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆) |
12 | 5, 11 | eqeltrd 2839 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∪ cuni 4912 ‘cfv 6563 SAlgcsalg 46264 SalGencsalgen 46268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-salg 46265 df-salgen 46269 |
This theorem is referenced by: salgensscntex 46300 |
Copyright terms: Public domain | W3C validator |