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Theorem issal 38287
 Description: Express the predicate " is a sigma-algebra." (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
issal SAlg
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem issal
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eleq2 2538 . . 3
2 raleq 2973 . . . 4
3 unieq 4198 . . . . . . 7
43difeq1d 3539 . . . . . 6
5 id 22 . . . . . 6
64, 5eleq12d 2543 . . . . 5
76ralbidv 2829 . . . 4
82, 7bitrd 261 . . 3
9 pweq 3945 . . . . 5
109raleqdv 2979 . . . 4
11 eleq2 2538 . . . . . 6
1211imbi2d 323 . . . . 5
1312ralbidv 2829 . . . 4
1410, 13bitrd 261 . . 3
151, 8, 143anbi123d 1365 . 2
16 df-salg 38282 . 2 SAlg
1715, 16elab2g 3175 1 SAlg
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   w3a 1007   wceq 1452   wcel 1904  wral 2756   cdif 3387  c0 3722  cpw 3942  cuni 4190   class class class wbr 4395  com 6711   cdom 7585  SAlgcsalg 38281 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-pw 3944  df-uni 4191  df-salg 38282 This theorem is referenced by:  pwsal  38288  salunicl  38289  saluncl  38290  prsal  38291  saldifcl  38292  0sal  38293  intsal  38301  issald  38304  caragensal  38465
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