Theorem sigagenss 30212

Description: The generated sigma-algebra is a subset of all sigma-algebras containing the generating set, i.e. the generated sigma-algebra is the smallest sigma-algebra containing the generating set, here A. (Contributed by Thierry Arnoux, 4-Jun-2017.)

Assertion

Ref	Expression
sigagenss	|- ( ( S e. (sigAlgebra ` U. A ) /\ A C_ S ) -> (sigaGen ` A ) C_ S )

Proof of Theorem sigagenss

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssexg 4804	. . . 4 |- ( ( A C_ S /\ S e. (sigAlgebra ` U. A ) ) -> A e. _V )
2	1	ancoms 469	. . 3 |- ( ( S e. (sigAlgebra ` U. A ) /\ A C_ S ) -> A e. _V )
3		sigagenval 30203	. . 3 |- ( A e. _V -> (sigaGen ` A ) = |^| { s e. (sigAlgebra ` U. A ) | A C_ s } )
4	2, 3	syl 17	. 2 |- ( ( S e. (sigAlgebra ` U. A ) /\ A C_ S ) -> (sigaGen ` A ) = |^| { s e. (sigAlgebra ` U. A ) | A C_ s } )
5		sseq2 3627	. . 3 |- ( s = S -> ( A C_ s <-> A C_ S ) )
6	5	intminss 4503	. 2 |- ( ( S e. (sigAlgebra ` U. A ) /\ A C_ S ) -> |^| { s e. (sigAlgebra ` U. A ) | A C_ s } C_ S )
7	4, 6	eqsstrd 3639	1 |- ( ( S e. (sigAlgebra ` U. A ) /\ A C_ S ) -> (sigaGen ` A ) C_ S )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   U.cuni 4436   |^|cint 4475   ` cfv 5888  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-iota 5851  df-fun 5890  df-fv 5896  df-siga 30171  df-sigagen 30202

This theorem is referenced by:  sigagenss2  30213  sigagenid  30214  imambfm  30324