Theorem elsigagen2 27973

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 996	. . 3 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> A e. V )
2	1	sgsiga 27967	. 2 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> (sigaGen ` A ) e. U. ran sigAlgebra )
3		sssigagen 27970	. . . 4 |- ( A e. V -> A C_ (sigaGen ` A ) )
4		sspwb 4702	. . . . 5 |- ( A C_ (sigaGen ` A ) <-> ~P A C_ ~P (sigaGen ` A ) )
5	4	biimpi 194	. . . 4 |- ( A C_ (sigaGen ` A ) -> ~P A C_ ~P (sigaGen ` A ) )
6	1, 3, 5	3syl 20	. . 3 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> ~P A C_ ~P (sigaGen ` A ) )
7		simp2 997	. . . 4 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> B C_ A )
8		simp3 998	. . . . 5 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> B ~<_ om )
9		ctex 27354	. . . . 5 |- ( B ~<_ om -> B e. _V )
10		elpwg 4024	. . . . 5 |- ( B e. _V -> ( B e. ~P A <-> B C_ A ) )
11	8, 9, 10	3syl 20	. . . 4 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> ( B e. ~P A <-> B C_ A ) )
12	7, 11	mpbird 232	. . 3 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> B e. ~P A )
13	6, 12	sseldd 3510	. 2 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> B e. ~P (sigaGen ` A ) )
14		sigaclcu 27942	. 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 1228	1 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> U. B e. (sigaGen ` A ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ w3a 973   e. wcel 1767   _Vcvv 3118   C_ wss 3481   ~Pcpw 4016   U.cuni 4251   class class class wbr 4453   ran crn 5006   ` cfv 5594   omcom 6695   ~<_ cdom 7526  sigAlgebracsiga 27932  sigaGencsigagen 27963

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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fv 5602  df-dom 7530  df-siga 27933  df-sigagen 27964

This theorem is referenced by:  sxbrsigalem1  28081