Theorem srossspw 30239

Description: A semi-ring of sets is a collection of subsets of O. (Contributed by Thierry Arnoux, 18-Jul-2020.)

Hypothesis

Ref	Expression
issros.1	|- N = { s e. ~P ~P O | ( (/) e. s /\ A. x e. s A. y e. s ( ( x i^i y ) e. s /\ E. z e. ~P s ( z e. Fin /\ Disj t e. z t /\ ( x \ y ) = U. z ) ) ) }

Assertion

Ref	Expression
srossspw	|- ( S e. N -> S C_ ~P O )

Distinct variable groups:   t, s, x, y   O, s   S, s, x, y, z

Allowed substitution hints:   S( t)   N( x, y, z, t, s)   O( x, y, z, t)

Proof of Theorem srossspw

Step	Hyp	Ref	Expression
1		issros.1	. . . 4 |- N = { s e. ~P ~P O | ( (/) e. s /\ A. x e. s A. y e. s ( ( x i^i y ) e. s /\ E. z e. ~P s ( z e. Fin /\ Disj t e. z t /\ ( x \ y ) = U. z ) ) ) }
2	1	issros 30238	. . 3 |- ( S e. N <-> ( S e. ~P ~P O /\ (/) e. S /\ A. x e. S A. y e. S ( ( x i^i y ) e. S /\ E. z e. ~P S ( z e. Fin /\ Disj t e. z t /\ ( x \ y ) = U. z ) ) ) )
3	2	simp1bi 1076	. 2 |- ( S e. N -> S e. ~P ~P O )
4	3	elpwid 4170	1 |- ( S e. N -> S C_ ~P O )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436  Disj wdisj 4620   Fincfn 7955

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160

This theorem is referenced by: (None)