Theorem baselsiga 27755

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 3122	. 2 |- ( S e. (sigAlgebra ` A ) -> S e. _V )
2		issiga 27751	. . . 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 1008	. 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 973   e. wcel 1767   A.wral 2814   _Vcvv 3113   \ cdif 3473   C_ wss 3476   ~Pcpw 4010   U.cuni 4245   class class class wbr 4447   ` cfv 5586   omcom 6678   ~<_ cdom 7511  sigAlgebracsiga 27747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-siga 27748

This theorem is referenced by:  unielsiga  27768  cldssbrsiga  27798  1stmbfm  27871  2ndmbfm  27872  unveldomd  27994  probmeasb  28009  dstrvprob  28050