Theorem pwsiga 28578

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 3461	. . 3 |- ~P O C_ ~P O
2	1	a1i 11	. 2 |- ( O e. V -> ~P O C_ ~P O )
3		pwidg 3968	. . 3 |- ( O e. V -> O e. ~P O )
4		difss 3570	. . . . . 6 |- ( O \ x ) C_ O
5		elpw2g 4557	. . . . . 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 2816	. . 3 |- ( O e. V -> A. x e. ~P O ( O \ x ) e. ~P O )
9		sspwuni 4360	. . . . . . . 8 |- ( x C_ ~P O <-> U. x C_ O )
10		vex 3062	. . . . . . . . . 10 |- x e. _V
11	10	uniex 6578	. . . . . . . . 9 |- U. x e. _V
12	11	elpw 3961	. . . . . . . 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 3964	. . . . . 6 |- ( x e. ~P ~P O -> x C_ ~P O )
17	16	imim1i 57	. . . . 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 2816	. . 3 |- ( O e. V -> A. x e. ~P ~P O ( x ~<_ om -> U. x e. ~P O ) )
20	3, 8, 19	3jca 1177	. 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 4578	. . 3 |- ( O e. V -> ~P O e. _V )
22		issiga 28559	. . 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 923	1 |- ( O e. V -> ~P O e. (sigAlgebra ` O ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   e. wcel 1842   A.wral 2754   _Vcvv 3059   \ cdif 3411   C_ wss 3414   ~Pcpw 3955   U.cuni 4191   class class class wbr 4395   ` cfv 5569   omcom 6683   ~<_ cdom 7552  sigAlgebracsiga 28555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-siga 28556

This theorem is referenced by:  sigagenval  28588  dmsigagen  28592  ldsysgenld  28608  pwcntmeas  28675  ddemeas  28685  mbfmcnt  28716