Theorem sigapisys 30218

Description: All sigma-algebras are pi-systems. (Contributed by Thierry Arnoux, 13-Jun-2020.)

Hypothesis

Ref	Expression
ispisys.p	|- P = { s e. ~P ~P O | ( fi ` s ) C_ s }

Assertion

Ref	Expression
sigapisys	|- (sigAlgebra ` O ) C_ P

Distinct variable group:   O, s

Allowed substitution hint:   P( s)

Proof of Theorem sigapisys

Dummy variables t x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sigasspw 30179	. . . . 5 |- ( t e. (sigAlgebra ` O ) -> t C_ ~P O )
2		selpw 4165	. . . . 5 |- ( t e. ~P ~P O <-> t C_ ~P O )
3	1, 2	sylibr 224	. . . 4 |- ( t e. (sigAlgebra ` O ) -> t e. ~P ~P O )
4		elrnsiga 30189	. . . . . . 7 |- ( t e. (sigAlgebra ` O ) -> t e. U. ran sigAlgebra )
5	4	adantr 481	. . . . . 6 |- ( ( t e. (sigAlgebra ` O ) /\ x e. ( ( ~P t i^i Fin ) \ { (/) } ) ) -> t e. U. ran sigAlgebra )
6		eldifsn 4317	. . . . . . . . . 10 |- ( x e. ( ( ~P t i^i Fin ) \ { (/) } ) <-> ( x e. ( ~P t i^i Fin ) /\ x =/= (/) ) )
7	6	biimpi 206	. . . . . . . . 9 |- ( x e. ( ( ~P t i^i Fin ) \ { (/) } ) -> ( x e. ( ~P t i^i Fin ) /\ x =/= (/) ) )
8	7	adantl 482	. . . . . . . 8 |- ( ( t e. (sigAlgebra ` O ) /\ x e. ( ( ~P t i^i Fin ) \ { (/) } ) ) -> ( x e. ( ~P t i^i Fin ) /\ x =/= (/) ) )
9	8	simpld 475	. . . . . . 7 |- ( ( t e. (sigAlgebra ` O ) /\ x e. ( ( ~P t i^i Fin ) \ { (/) } ) ) -> x e. ( ~P t i^i Fin ) )
10	9	elin1d 3802	. . . . . 6 |- ( ( t e. (sigAlgebra ` O ) /\ x e. ( ( ~P t i^i Fin ) \ { (/) } ) ) -> x e. ~P t )
11	9	elin2d 3803	. . . . . . 7 |- ( ( t e. (sigAlgebra ` O ) /\ x e. ( ( ~P t i^i Fin ) \ { (/) } ) ) -> x e. Fin )
12		fict 8550	. . . . . . 7 |- ( x e. Fin -> x ~<_ om )
13	11, 12	syl 17	. . . . . 6 |- ( ( t e. (sigAlgebra ` O ) /\ x e. ( ( ~P t i^i Fin ) \ { (/) } ) ) -> x ~<_ om )
14	8	simprd 479	. . . . . 6 |- ( ( t e. (sigAlgebra ` O ) /\ x e. ( ( ~P t i^i Fin ) \ { (/) } ) ) -> x =/= (/) )
15		sigaclci 30195	. . . . . 6 |- ( ( ( t e. U. ran sigAlgebra /\ x e. ~P t ) /\ ( x ~<_ om /\ x =/= (/) ) ) -> |^| x e. t )
16	5, 10, 13, 14, 15	syl22anc 1327	. . . . 5 |- ( ( t e. (sigAlgebra ` O ) /\ x e. ( ( ~P t i^i Fin ) \ { (/) } ) ) -> |^| x e. t )
17	16	ralrimiva 2966	. . . 4 |- ( t e. (sigAlgebra ` O ) -> A. x e. ( ( ~P t i^i Fin ) \ { (/) } ) |^| x e. t )
18	3, 17	jca 554	. . 3 |- ( t e. (sigAlgebra ` O ) -> ( t e. ~P ~P O /\ A. x e. ( ( ~P t i^i Fin ) \ { (/) } ) |^| x e. t ) )
19		ispisys.p	. . . 4 |- P = { s e. ~P ~P O | ( fi ` s ) C_ s }
20	19	ispisys2 30216	. . 3 |- ( t e. P <-> ( t e. ~P ~P O /\ A. x e. ( ( ~P t i^i Fin ) \ { (/) } ) |^| x e. t ) )
21	18, 20	sylibr 224	. 2 |- ( t e. (sigAlgebra ` O ) -> t e. P )
22	21	ssriv 3607	1 |- (sigAlgebra ` O ) C_ P

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   |^|cint 4475   class class class wbr 4653   ran crn 5115   ` cfv 5888   omcom 7065   ~<_ cdom 7953   Fincfn 7955   ficfi 8316  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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-card 8765  df-acn 8768  df-ac 8939  df-siga 30171

This theorem is referenced by:  sigapildsys  30225