Theorem pwsiga 27767

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 3523	. . 3 |- ~P O C_ ~P O
2	1	a1i 11	. 2 |- ( O e. V -> ~P O C_ ~P O )
3		pwidg 4023	. . 3 |- ( O e. V -> O e. ~P O )
4		difss 3631	. . . . . 6 |- ( O \ x ) C_ O
5		elpw2g 4610	. . . . . 6 |- ( O e. V -> ( ( O \ x ) e. ~P O <-> ( O \ x ) C_ O ) )
6	4, 5	mpbiri 233	. . . . 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 2876	. . 3 |- ( O e. V -> A. x e. ~P O ( O \ x ) e. ~P O )
9		sspwuni 4411	. . . . . . . 8 |- ( x C_ ~P O <-> U. x C_ O )
10		vex 3116	. . . . . . . . . 10 |- x e. _V
11	10	uniex 6578	. . . . . . . . 9 |- U. x e. _V
12	11	elpw 4016	. . . . . . . 8 |- ( U. x e. ~P O <-> U. x C_ O )
13	9, 12	bitr4i 252	. . . . . . 7 |- ( x C_ ~P O <-> U. x e. ~P O )
14	13	biimpi 194	. . . . . 6 |- ( x C_ ~P O -> U. x e. ~P O )
15	14	a1d 25	. . . . 5 |- ( x C_ ~P O -> ( x ~<_ om -> U. x e. ~P O ) )
16		elpwi 4019	. . . . . 6 |- ( x e. ~P ~P O -> x C_ ~P O )
17	16	imim1i 58	. . . . 5 |- ( ( x C_ ~P O -> ( x ~<_ om -> U. x e. ~P O ) ) -> ( x e. ~P ~P O -> ( x ~<_ om -> U. x e. ~P O ) ) )
18	15, 17	mp1i 12	. . . 4 |- ( O e. V -> ( x e. ~P ~P O -> ( x ~<_ om -> U. x e. ~P O ) ) )
19	18	ralrimiv 2876	. . 3 |- ( O e. V -> A. x e. ~P ~P O ( x ~<_ om -> U. x e. ~P O ) )
20	3, 8, 19	3jca 1176	. 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 ) ) )
21		pwexg 4631	. . 3 |- ( O e. V -> ~P O e. _V )
22		issiga 27748	. . 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 ) ) ) ) )
23	21, 22	syl 16	. 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 ) ) ) ) )
24	2, 20, 23	mpbir2and 920	1 |- ( O e. V -> ~P O e. (sigAlgebra ` O ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   e. wcel 1767   A.wral 2814   _Vcvv 3113   \ cdif 3473   C_ wss 3476   ~Pcpw 4010   U.cuni 4245   class class class wbr 4447   ` cfv 5586   omcom 6678   ~<_ cdom 7511  sigAlgebracsiga 27744

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 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-siga 27745

This theorem is referenced by:  sigagenval  27777  dmsigagen  27781  pwcntmeas  27835  ddemeas  27845  mbfmcnt  27876