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.

Step	Hyp	Ref	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 ) ) ) ) )
3	2	simplbda 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 ) ) )
4	3	simp1d 1000	. 2 |- ( ( S e. _V /\ S e. (sigAlgebra ` A ) ) -> A e. S )
5	1, 4	mpancom 669	1 |- ( S e. (sigAlgebra ` A ) -> A e. S )

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