Theorem elsigagen2 30211

Description: Any countable union of elements of a set is also in the sigma-algebra that set generates. (Contributed by Thierry Arnoux, 17-Sep-2017.)

Assertion

Ref	Expression
elsigagen2	|- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> U. B e. (sigaGen ` A ) )

Proof of Theorem elsigagen2

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> A e. V )
2	1	sgsiga 30205	. 2 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> (sigaGen ` A ) e. U. ran sigAlgebra )
3		sssigagen 30208	. . . 4 |- ( A e. V -> A C_ (sigaGen ` A ) )
4		sspwb 4917	. . . . 5 |- ( A C_ (sigaGen ` A ) <-> ~P A C_ ~P (sigaGen ` A ) )
5	4	biimpi 206	. . . 4 |- ( A C_ (sigaGen ` A ) -> ~P A C_ ~P (sigaGen ` A ) )
6	1, 3, 5	3syl 18	. . 3 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> ~P A C_ ~P (sigaGen ` A ) )
7		simp2 1062	. . . 4 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> B C_ A )
8		simp3 1063	. . . . 5 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> B ~<_ om )
9		ctex 7970	. . . . 5 |- ( B ~<_ om -> B e. _V )
10		elpwg 4166	. . . . 5 |- ( B e. _V -> ( B e. ~P A <-> B C_ A ) )
11	8, 9, 10	3syl 18	. . . 4 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> ( B e. ~P A <-> B C_ A ) )
12	7, 11	mpbird 247	. . 3 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> B e. ~P A )
13	6, 12	sseldd 3604	. 2 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> B e. ~P (sigaGen ` A ) )
14		sigaclcu 30180	. 2 |- ( ( (sigaGen ` A ) e. U. ran sigAlgebra /\ B e. ~P (sigaGen ` A ) /\ B ~<_ om ) -> U. B e. (sigaGen ` A ) )
15	2, 13, 8, 14	syl3anc 1326	1 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> U. B e. (sigaGen ` A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   ran crn 5115   ` cfv 5888   omcom 7065   ~<_ cdom 7953  sigAlgebracsiga 30170  sigaGencsigagen 30201

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  ax-un 6949

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-int 4476  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896  df-dom 7957  df-siga 30171  df-sigagen 30202

This theorem is referenced by:  sxbrsigalem1  30347