Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  baselsiga Structured version   Unicode version

Theorem baselsiga 26696
Description: A sigma-algebra contains its base universe set. (Contributed by Thierry Arnoux, 26-Oct-2016.)
Assertion
Ref Expression
baselsiga  |-  ( S  e.  (sigAlgebra `  A )  ->  A  e.  S )

Proof of Theorem baselsiga
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 3080 . 2  |-  ( S  e.  (sigAlgebra `  A )  ->  S  e.  _V )
2 issiga 26692 . . . 4  |-  ( S  e.  _V  ->  ( S  e.  (sigAlgebra `  A
)  <->  ( S  C_  ~P A  /\  ( A  e.  S  /\  A. x  e.  S  ( A  \  x )  e.  S  /\  A. x  e.  ~P  S
( x  ~<_  om  ->  U. x  e.  S ) ) ) ) )
32simplbda 624 . . 3  |-  ( ( S  e.  _V  /\  S  e.  (sigAlgebra `  A
) )  ->  ( A  e.  S  /\  A. x  e.  S  ( A  \  x )  e.  S  /\  A. x  e.  ~P  S
( x  ~<_  om  ->  U. x  e.  S ) ) )
43simp1d 1000 . 2  |-  ( ( S  e.  _V  /\  S  e.  (sigAlgebra `  A
) )  ->  A  e.  S )
51, 4mpancom 669 1  |-  ( S  e.  (sigAlgebra `  A )  ->  A  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1758   A.wral 2795   _Vcvv 3071    \ cdif 3426    C_ wss 3429   ~Pcpw 3961   U.cuni 4192   class class class wbr 4393   ` cfv 5519   omcom 6579    ~<_ cdom 7411  sigAlgebracsiga 26688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-iota 5482  df-fun 5521  df-fv 5527  df-siga 26689
This theorem is referenced by:  unielsiga  26709  cldssbrsiga  26739  1stmbfm  26812  2ndmbfm  26813  unveldomd  26935  probmeasb  26950  dstrvprob  26991
  Copyright terms: Public domain W3C validator