Theorem 0elros 30233

Description: A ring of sets contains the empty set. (Contributed by Thierry Arnoux, 18-Jul-2020.)

Hypothesis

Ref	Expression
isros.1	|- Q = { s e. ~P ~P O | ( (/) e. s /\ A. x e. s A. y e. s ( ( x u. y ) e. s /\ ( x \ y ) e. s ) ) }

Assertion

Ref	Expression
0elros	|- ( S e. Q -> (/) e. S )

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

Allowed substitution hints:   Q( x, y, s)   O( x, y)

Proof of Theorem 0elros

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isros.1	. . 3 |- Q = { s e. ~P ~P O | ( (/) e. s /\ A. x e. s A. y e. s ( ( x u. y ) e. s /\ ( x \ y ) e. s ) ) }
2	1	isros 30231	. 2 |- ( S e. Q <-> ( S e. ~P ~P O /\ (/) e. S /\ A. u e. S A. v e. S ( ( u u. v ) e. S /\ ( u \ v ) e. S ) ) )
3	2	simp2bi 1077	1 |- ( S e. Q -> (/) e. S )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ cdif 3571   u. cun 3572   (/)c0 3915   ~Pcpw 4158

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-rab 2921  df-v 3202  df-dif 3577  df-un 3579

This theorem is referenced by:  fiunelros  30237  rossros  30243