Theorem sigasspw 30179

Description: A sigma-algebra is a set of subset of the base set. (Contributed by Thierry Arnoux, 18-Jan-2017.)

Assertion

Ref	Expression
sigasspw	|- ( S e. (sigAlgebra ` A ) -> S C_ ~P A )

Proof of Theorem sigasspw

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. . 3 |- ( S e. (sigAlgebra ` A ) -> S e. _V )
2		issiga 30174	. . . 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	biimpa 501	. . 3 |- ( ( 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 ) ) ) )
4	1, 3	mpancom 703	. 2 |- ( 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 ) ) ) )
5	4	simpld 475	1 |- ( S e. (sigAlgebra ` A ) -> S C_ ~P A )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   ` cfv 5888   omcom 7065   ~<_ cdom 7953  sigAlgebracsiga 30170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-siga 30171

This theorem is referenced by:  elsigass  30188  insiga  30200  sigapisys  30218  sigaldsys  30222  brsigasspwrn  30248  1stmbfm  30322  2ndmbfm  30323