Theorem elsigagen2 26759

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 988	. . 3 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> A e. V )
2	1	sgsiga 26753	. 2 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> (sigaGen ` A ) e. U. ran sigAlgebra )
3		sssigagen 26756	. . . 4 |- ( A e. V -> A C_ (sigaGen ` A ) )
4		sspwb 4652	. . . . 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 989	. . . 4 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> B C_ A )
8		simp3 990	. . . . 5 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> B ~<_ om )
9		ctex 26186	. . . . 5 |- ( B ~<_ om -> B e. _V )
10		elpwg 3979	. . . . 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 3468	. 2 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> B e. ~P (sigaGen ` A ) )
14		sigaclcu 26728	. 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 1219	1 |- ( ( A e. V /\ B C_ A /\ B ~<_ om ) -> U. B e. (sigaGen ` A ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ w3a 965   e. wcel 1758   _Vcvv 3078   C_ wss 3439   ~Pcpw 3971   U.cuni 4202   class class class wbr 4403   ran crn 4952   ` cfv 5529   omcom 6589   ~<_ cdom 7421  sigAlgebracsiga 26718  sigaGencsigagen 26749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fv 5537  df-dom 7425  df-siga 26719  df-sigagen 26750

This theorem is referenced by:  sxbrsigalem1  26867