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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 Disj
Assertion
Ref Expression
isldsys Disj
Distinct variable groups:   ,   ,,   ,,
Allowed substitution hints:   ()   (,,)   ()

Proof of Theorem isldsys
StepHypRef Expression
1 eleq2 2538 . . 3
2 eleq2 2538 . . . 4
32raleqbi1dv 2981 . . 3
4 pweq 3945 . . . 4
5 eleq2 2538 . . . . 5
65imbi2d 323 . . . 4 Disj Disj
74, 6raleqbidv 2987 . . 3 Disj Disj
81, 3, 73anbi123d 1365 . 2 Disj Disj
9 isldsys.l . 2 Disj
108, 9elrab2 3186 1 Disj
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   w3a 1007   wceq 1452   wcel 1904  wral 2756  crab 2760   cdif 3387  c0 3722  cpw 3942  cuni 4190  Disj wdisj 4366   class class class wbr 4395  com 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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