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Mathbox for Glauco Siliprandi |
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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 44826 | . . 3 ⊢ (𝜑 → ∪ 𝑆 = 𝑈) |
5 | 4 | eqcomd 2737 | . 2 ⊢ (𝜑 → 𝑈 = ∪ 𝑆) |
6 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑆 = (SalGen‘𝑋)) |
7 | salgencl 44821 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg) | |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (SalGen‘𝑋) ∈ SAlg) |
9 | 6, 8 | eqeltrd 2832 | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
10 | saluni 44814 | . . 3 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆) |
12 | 5, 11 | eqeltrd 2832 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∪ cuni 4901 ‘cfv 6532 SAlgcsalg 44797 SalGencsalgen 44801 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 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-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6484 df-fun 6534 df-fv 6540 df-salg 44798 df-salgen 44802 |
This theorem is referenced by: salgensscntex 44833 |
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