Theorem pwsiga 26578

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 3380	. . 3 |- ~P O C_ ~P O
2	1	a1i 11	. 2 |- ( O e. V -> ~P O C_ ~P O )
3		pwidg 3878	. . 3 |- ( O e. V -> O e. ~P O )
4		difss 3488	. . . . . 6 |- ( O \ x ) C_ O
5		elpw2g 4460	. . . . . 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 2803	. . 3 |- ( O e. V -> A. x e. ~P O ( O \ x ) e. ~P O )
9		sspwuni 4261	. . . . . . . 8 |- ( x C_ ~P O <-> U. x C_ O )
10		vex 2980	. . . . . . . . . 10 |- x e. _V
11	10	uniex 6381	. . . . . . . . 9 |- U. x e. _V
12	11	elpw 3871	. . . . . . . 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 3874	. . . . . 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 2803	. . 3 |- ( O e. V -> A. x e. ~P ~P O ( x ~<_ om -> U. x e. ~P O ) )
20	3, 8, 19	3jca 1168	. 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 4481	. . 3 |- ( O e. V -> ~P O e. _V )
22		issiga 26559	. . 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 913	1 |- ( O e. V -> ~P O e. (sigAlgebra ` O ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   e. wcel 1756   A.wral 2720   _Vcvv 2977   \ cdif 3330   C_ wss 3333   ~Pcpw 3865   U.cuni 4096   class class class wbr 4297   ` cfv 5423   omcom 6481   ~<_ cdom 7313  sigAlgebracsiga 26555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-siga 26556

This theorem is referenced by:  sigagenval  26588  dmsigagen  26592  pwcntmeas  26646  ddemeas  26657  mbfmcnt  26688