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Theorem isldsys 29052
Description: The property of being a lambda-system or Dynkin system. Lambda-systems contain the empty set, are closed under complement, and closed under countable disjoint union. (Contributed by Thierry Arnoux, 13-Jun-2020.)
Hypothesis
Ref Expression
isldsys.l  |-  L  =  { s  e.  ~P ~P O  |  ( (/) 
e.  s  /\  A. x  e.  s  ( O  \  x )  e.  s  /\  A. x  e.  ~P  s ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  ->  U. x  e.  s
) ) }
Assertion
Ref Expression
isldsys  |-  ( S  e.  L  <->  ( S  e.  ~P ~P O  /\  ( (/)  e.  S  /\  A. x  e.  S  ( O  \  x )  e.  S  /\  A. x  e.  ~P  S
( ( x  ~<_  om 
/\ Disj  y  e.  x  y )  ->  U. x  e.  S ) ) ) )
Distinct variable groups:    y, s    O, s, x    S, s, x
Allowed substitution hints:    S( y)    L( x, y, s)    O( y)

Proof of Theorem isldsys
StepHypRef Expression
1 eleq2 2538 . . 3  |-  ( s  =  S  ->  ( (/) 
e.  s  <->  (/)  e.  S
) )
2 eleq2 2538 . . . 4  |-  ( s  =  S  ->  (
( O  \  x
)  e.  s  <->  ( O  \  x )  e.  S
) )
32raleqbi1dv 2981 . . 3  |-  ( s  =  S  ->  ( A. x  e.  s 
( O  \  x
)  e.  s  <->  A. x  e.  S  ( O  \  x )  e.  S
) )
4 pweq 3945 . . . 4  |-  ( s  =  S  ->  ~P s  =  ~P S
)
5 eleq2 2538 . . . . 5  |-  ( s  =  S  ->  ( U. x  e.  s  <->  U. x  e.  S ) )
65imbi2d 323 . . . 4  |-  ( s  =  S  ->  (
( ( x  ~<_  om 
/\ Disj  y  e.  x  y )  ->  U. x  e.  s )  <->  ( (
x  ~<_  om  /\ Disj  y  e.  x  y )  ->  U. x  e.  S
) ) )
74, 6raleqbidv 2987 . . 3  |-  ( s  =  S  ->  ( A. x  e.  ~P  s ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  ->  U. x  e.  s )  <->  A. x  e.  ~P  S ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  ->  U. x  e.  S
) ) )
81, 3, 73anbi123d 1365 . 2  |-  ( s  =  S  ->  (
( (/)  e.  s  /\  A. x  e.  s  ( O  \  x )  e.  s  /\  A. x  e.  ~P  s
( ( x  ~<_  om 
/\ Disj  y  e.  x  y )  ->  U. x  e.  s ) )  <->  ( (/)  e.  S  /\  A. x  e.  S  ( O  \  x
)  e.  S  /\  A. x  e.  ~P  S
( ( x  ~<_  om 
/\ Disj  y  e.  x  y )  ->  U. x  e.  S ) ) ) )
9 isldsys.l . 2  |-  L  =  { s  e.  ~P ~P O  |  ( (/) 
e.  s  /\  A. x  e.  s  ( O  \  x )  e.  s  /\  A. x  e.  ~P  s ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  ->  U. x  e.  s
) ) }
108, 9elrab2 3186 1  |-  ( S  e.  L  <->  ( S  e.  ~P ~P O  /\  ( (/)  e.  S  /\  A. x  e.  S  ( O  \  x )  e.  S  /\  A. x  e.  ~P  S
( ( x  ~<_  om 
/\ Disj  y  e.  x  y )  ->  U. x  e.  S ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   {crab 2760    \ cdif 3387   (/)c0 3722   ~Pcpw 3942   U.cuni 4190  Disj wdisj 4366   class class class wbr 4395   omcom 6711    ~<_ cdom 7585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rab 2765  df-v 3033  df-in 3397  df-ss 3404  df-pw 3944
This theorem is referenced by:  pwldsys  29053  unelldsys  29054  sigaldsys  29055  ldsysgenld  29056  sigapildsyslem  29057  sigapildsys  29058  ldgenpisyslem1  29059
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