Theorem pwsiga 28945

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 3450	. . 3 |- ~P O C_ ~P O
2	1	a1i 11	. 2 |- ( O e. V -> ~P O C_ ~P O )
3		pwidg 3963	. . 3 |- ( O e. V -> O e. ~P O )
4		difss 3559	. . . . . 6 |- ( O \ x ) C_ O
5		elpw2g 4565	. . . . . 6 |- ( O e. V -> ( ( O \ x ) e. ~P O <-> ( O \ x ) C_ O ) )
6	4, 5	mpbiri 237	. . . . 5 |- ( O e. V -> ( O \ x ) e. ~P O )
7	6	a1d 26	. . . 4 |- ( O e. V -> ( x e. ~P O -> ( O \ x ) e. ~P O ) )
8	7	ralrimiv 2799	. . 3 |- ( O e. V -> A. x e. ~P O ( O \ x ) e. ~P O )
9		sspwuni 4366	. . . . . . . 8 |- ( x C_ ~P O <-> U. x C_ O )
10		vex 3047	. . . . . . . . . 10 |- x e. _V
11	10	uniex 6584	. . . . . . . . 9 |- U. x e. _V
12	11	elpw 3956	. . . . . . . 8 |- ( U. x e. ~P O <-> U. x C_ O )
13	9, 12	bitr4i 256	. . . . . . 7 |- ( x C_ ~P O <-> U. x e. ~P O )
14	13	biimpi 198	. . . . . 6 |- ( x C_ ~P O -> U. x e. ~P O )
15	14	a1d 26	. . . . 5 |- ( x C_ ~P O -> ( x ~<_ om -> U. x e. ~P O ) )
16		elpwi 3959	. . . . . 6 |- ( x e. ~P ~P O -> x C_ ~P O )
17	16	imim1i 60	. . . . 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 13	. . . 4 |- ( O e. V -> ( x e. ~P ~P O -> ( x ~<_ om -> U. x e. ~P O ) ) )
19	18	ralrimiv 2799	. . 3 |- ( O e. V -> A. x e. ~P ~P O ( x ~<_ om -> U. x e. ~P O ) )
20	3, 8, 19	3jca 1187	. 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 4586	. . 3 |- ( O e. V -> ~P O e. _V )
22		issiga 28926	. . 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 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 ) ) ) ) )
24	2, 20, 23	mpbir2and 932	1 |- ( O e. V -> ~P O e. (sigAlgebra ` O ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984   e. wcel 1886   A.wral 2736   _Vcvv 3044   \ cdif 3400   C_ wss 3403   ~Pcpw 3950   U.cuni 4197   class class class wbr 4401   ` cfv 5581   omcom 6689   ~<_ cdom 7564  sigAlgebracsiga 28922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-iota 5545  df-fun 5583  df-fv 5589  df-siga 28923

This theorem is referenced by:  sigagenval  28955  dmsigagen  28959  ldsysgenld  28975  pwcntmeas  29042  ddemeas  29052  mbfmcnt  29083