Theorem baselsiga 26510

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 2976	. 2 |- ( S e. (sigAlgebra ` A ) -> S e. _V )
2		issiga 26506	. . . 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 1756   A.wral 2710   _Vcvv 2967   \ cdif 3320   C_ wss 3323   ~Pcpw 3855   U.cuni 4086   class class class wbr 4287   ` cfv 5413   omcom 6471   ~<_ cdom 7300  sigAlgebracsiga 26502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-iota 5376  df-fun 5415  df-fv 5421  df-siga 26503

This theorem is referenced by:  unielsiga  26523  cldssbrsiga  26553  1stmbfm  26627  2ndmbfm  26628  unveldomd  26750  probmeasb  26765  dstrvprob  26806