Theorem pwsiga 30193

Description: Any power set forms a sigma-algebra. (Contributed by Thierry Arnoux, 13-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)

Assertion

Ref	Expression
pwsiga	|- ( O e. V -> ~P O e. (sigAlgebra ` O ) )

Proof of Theorem pwsiga

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3624	. . 3 |- ~P O C_ ~P O
2	1	a1i 11	. 2 |- ( O e. V -> ~P O C_ ~P O )
3		pwidg 4173	. . 3 |- ( O e. V -> O e. ~P O )
4		difss 3737	. . . . . 6 |- ( O \ x ) C_ O
5		elpw2g 4827	. . . . . 6 |- ( O e. V -> ( ( O \ x ) e. ~P O <-> ( O \ x ) C_ O ) )
6	4, 5	mpbiri 248	. . . . 5 |- ( O e. V -> ( O \ x ) e. ~P O )
7	6	a1d 25	. . . 4 |- ( O e. V -> ( x e. ~P O -> ( O \ x ) e. ~P O ) )
8	7	ralrimiv 2965	. . 3 |- ( O e. V -> A. x e. ~P O ( O \ x ) e. ~P O )
9		sspwuni 4611	. . . . . . . 8 |- ( x C_ ~P O <-> U. x C_ O )
10		vuniex 6954	. . . . . . . . 9 |- U. x e. _V
11	10	elpw 4164	. . . . . . . 8 |- ( U. x e. ~P O <-> U. x C_ O )
12	9, 11	bitr4i 267	. . . . . . 7 |- ( x C_ ~P O <-> U. x e. ~P O )
13	12	biimpi 206	. . . . . 6 |- ( x C_ ~P O -> U. x e. ~P O )
14	13	a1d 25	. . . . 5 |- ( x C_ ~P O -> ( x ~<_ om -> U. x e. ~P O ) )
15		elpwi 4168	. . . . . 6 |- ( x e. ~P ~P O -> x C_ ~P O )
16	15	imim1i 63	. . . . 5 |- ( ( x C_ ~P O -> ( x ~<_ om -> U. x e. ~P O ) ) -> ( x e. ~P ~P O -> ( x ~<_ om -> U. x e. ~P O ) ) )
17	14, 16	mp1i 13	. . . 4 |- ( O e. V -> ( x e. ~P ~P O -> ( x ~<_ om -> U. x e. ~P O ) ) )
18	17	ralrimiv 2965	. . 3 |- ( O e. V -> A. x e. ~P ~P O ( x ~<_ om -> U. x e. ~P O ) )
19	3, 8, 18	3jca 1242	. 2 |- ( O e. V -> ( O e. ~P O /\ A. x e. ~P O ( O \ x ) e. ~P O /\ A. x e. ~P ~P O ( x ~<_ om -> U. x e. ~P O ) ) )
20		pwexg 4850	. . 3 |- ( O e. V -> ~P O e. _V )
21		issiga 30174	. . 3 |- ( ~P O e. _V -> ( ~P O e. (sigAlgebra ` O ) <-> ( ~P O C_ ~P O /\ ( O e. ~P O /\ A. x e. ~P O ( O \ x ) e. ~P O /\ A. x e. ~P ~P O ( x ~<_ om -> U. x e. ~P O ) ) ) ) )
22	20, 21	syl 17	. 2 |- ( O e. V -> ( ~P O e. (sigAlgebra ` O ) <-> ( ~P O C_ ~P O /\ ( O e. ~P O /\ A. x e. ~P O ( O \ x ) e. ~P O /\ A. x e. ~P ~P O ( x ~<_ om -> U. x e. ~P O ) ) ) ) )
23	2, 19, 22	mpbir2and 957	1 |- ( O e. V -> ~P O e. (sigAlgebra ` O ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ 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  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-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:  sigagenval  30203  dmsigagen  30207  ldsysgenld  30223  pwcntmeas  30290  ddemeas  30299  mbfmcnt  30330