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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > unisalgen2 | Structured version Visualization version GIF version |
Description: The union of a set belongs is equal to the union of the sigma-algebra generated by the set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
unisalgen2.x | β’ (π β π΄ β π) |
unisalgen2.s | β’ π = (SalGenβπ΄) |
Ref | Expression |
---|---|
unisalgen2 | β’ (π β βͺ π = βͺ π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unisalgen2.s | . . . . . 6 β’ π = (SalGenβπ΄) | |
2 | 1 | eqcomi 2735 | . . . . 5 β’ (SalGenβπ΄) = π |
3 | 2 | a1i 11 | . . . 4 β’ (π β (SalGenβπ΄) = π) |
4 | unisalgen2.x | . . . . 5 β’ (π β π΄ β π) | |
5 | 4 | dfsalgen2 45634 | . . . 4 β’ (π β ((SalGenβπ΄) = π β ((π β SAlg β§ βͺ π = βͺ π΄ β§ π΄ β π) β§ βπ₯ β SAlg ((βͺ π₯ = βͺ π΄ β§ π΄ β π₯) β π β π₯)))) |
6 | 3, 5 | mpbid 231 | . . 3 β’ (π β ((π β SAlg β§ βͺ π = βͺ π΄ β§ π΄ β π) β§ βπ₯ β SAlg ((βͺ π₯ = βͺ π΄ β§ π΄ β π₯) β π β π₯))) |
7 | 6 | simpld 494 | . 2 β’ (π β (π β SAlg β§ βͺ π = βͺ π΄ β§ π΄ β π)) |
8 | 7 | simp2d 1140 | 1 β’ (π β βͺ π = βͺ π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 β wss 3943 βͺ cuni 4902 βcfv 6537 SAlgcsalg 45601 SalGencsalgen 45605 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fv 6545 df-salg 45602 df-salgen 45606 |
This theorem is referenced by: incsmf 46035 decsmf 46060 |
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