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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 45599 | . . 3 β’ (π β βͺ π = π) |
5 | 4 | eqcomd 2730 | . 2 β’ (π β π = βͺ π) |
6 | 2 | a1i 11 | . . . 4 β’ (π β π = (SalGenβπ)) |
7 | salgencl 45594 | . . . . 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 2825 | . . 3 β’ (π β π β SAlg) |
10 | saluni 45587 | . . 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 2825 | 1 β’ (π β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βͺ cuni 4900 βcfv 6534 SAlgcsalg 45570 SalGencsalgen 45574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-iota 6486 df-fun 6536 df-fv 6542 df-salg 45571 df-salgen 45575 |
This theorem is referenced by: salgensscntex 45606 |
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