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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 | β’ (π β π β π) |
unisalgen.s | β’ π = (SalGenβπ) |
unisalgen.u | β’ π = βͺ π |
Ref | Expression |
---|---|
unisalgen | β’ (π β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unisalgen.x | . . . 4 β’ (π β π β π) | |
2 | unisalgen.s | . . . 4 β’ π = (SalGenβπ) | |
3 | unisalgen.u | . . . 4 β’ π = βͺ π | |
4 | 1, 2, 3 | salgenuni 44668 | . . 3 β’ (π β βͺ π = π) |
5 | 4 | eqcomd 2738 | . 2 β’ (π β π = βͺ π) |
6 | 2 | a1i 11 | . . . 4 β’ (π β π = (SalGenβπ)) |
7 | salgencl 44663 | . . . . 5 β’ (π β π β (SalGenβπ) β SAlg) | |
8 | 1, 7 | syl 17 | . . . 4 β’ (π β (SalGenβπ) β SAlg) |
9 | 6, 8 | eqeltrd 2833 | . . 3 β’ (π β π β SAlg) |
10 | saluni 44656 | . . 3 β’ (π β SAlg β βͺ π β π) | |
11 | 9, 10 | syl 17 | . 2 β’ (π β βͺ π β π) |
12 | 5, 11 | eqeltrd 2833 | 1 β’ (π β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βͺ cuni 4869 βcfv 6500 SAlgcsalg 44639 SalGencsalgen 44643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2703 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-salg 44640 df-salgen 44644 |
This theorem is referenced by: salgensscntex 44675 |
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