Theorem baselsiga 28549

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 3067	. 2 |- ( S e. (sigAlgebra ` A ) -> S e. _V )
2		issiga 28545	. . . 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 622	. . 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 1009	. 2 |- ( ( S e. _V /\ S e. (sigAlgebra ` A ) ) -> A e. S )
5	1, 4	mpancom 667	1 |- ( S e. (sigAlgebra ` A ) -> A e. S )

Syntax hints:   -> wi 4   /\ wa 367   /\ w3a 974   e. wcel 1842   A.wral 2753   _Vcvv 3058   \ cdif 3410   C_ wss 3413   ~Pcpw 3954   U.cuni 4190   class class class wbr 4394   ` cfv 5568   omcom 6682   ~<_ cdom 7551  sigAlgebracsiga 28541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-siga 28542

This theorem is referenced by:  unielsiga  28562  sigaldsys  28593  cldssbrsiga  28621  1stmbfm  28694  2ndmbfm  28695  unveldomd  28846  probmeasb  28861  dstrvprob  28902