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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 2742 | . . . . 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 44668 | . . . 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 496 | . 2 β’ (π β (π β SAlg β§ βͺ π = βͺ π΄ β§ π΄ β π)) |
8 | 7 | simp2d 1144 | 1 β’ (π β βͺ π = βͺ π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 β wss 3911 βͺ cuni 4866 βcfv 6497 SAlgcsalg 44635 SalGencsalgen 44639 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-salg 44636 df-salgen 44640 |
This theorem is referenced by: incsmf 45069 decsmf 45094 |
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