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 42614 | . . 3 ⊢ (𝜑 → ∪ 𝑆 = 𝑈) |
5 | 4 | eqcomd 2827 | . 2 ⊢ (𝜑 → 𝑈 = ∪ 𝑆) |
6 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑆 = (SalGen‘𝑋)) |
7 | salgencl 42609 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg) | |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (SalGen‘𝑋) ∈ SAlg) |
9 | 6, 8 | eqeltrd 2913 | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
10 | saluni 42603 | . . 3 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆) |
12 | 5, 11 | eqeltrd 2913 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∪ cuni 4831 ‘cfv 6349 SAlgcsalg 42587 SalGencsalgen 42591 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4869 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-salg 42588 df-salgen 42592 |
This theorem is referenced by: salgensscntex 42621 |
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