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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 46352 | . . 3 ⊢ (𝜑 → ∪ 𝑆 = 𝑈) | 
| 5 | 4 | eqcomd 2743 | . 2 ⊢ (𝜑 → 𝑈 = ∪ 𝑆) | 
| 6 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑆 = (SalGen‘𝑋)) | 
| 7 | salgencl 46347 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg) | |
| 8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (SalGen‘𝑋) ∈ SAlg) | 
| 9 | 6, 8 | eqeltrd 2841 | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | 
| 10 | saluni 46340 | . . 3 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆) | |
| 11 | 9, 10 | syl 17 | . 2 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆) | 
| 12 | 5, 11 | eqeltrd 2841 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∪ cuni 4907 ‘cfv 6561 SAlgcsalg 46323 SalGencsalgen 46327 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-salg 46324 df-salgen 46328 | 
| This theorem is referenced by: salgensscntex 46359 | 
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