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 40555 | . . 3 ⊢ (𝜑 → ∪ 𝑆 = 𝑈) |
5 | 4 | eqcomd 2628 | . 2 ⊢ (𝜑 → 𝑈 = ∪ 𝑆) |
6 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑆 = (SalGen‘𝑋)) |
7 | salgencl 40550 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg) | |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (SalGen‘𝑋) ∈ SAlg) |
9 | 6, 8 | eqeltrd 2701 | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
10 | saluni 40544 | . . 3 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆) |
12 | 5, 11 | eqeltrd 2701 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∪ cuni 4436 ‘cfv 5888 SAlgcsalg 40528 SalGencsalgen 40532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-salg 40529 df-salgen 40533 |
This theorem is referenced by: salgensscntex 40562 |
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