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Definition df-iord 4130
Description: Define the ordinal predicate, which is true for a class that is transitive and whose elements are transitive. Definition of ordinal in [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". (Contributed by Jim Kingdon, 10-Oct-2018.) Use its alias dford3 4131 instead for naming consistency with set.mm. (New usage is discouraged.)
Assertion
Ref Expression
df-iord  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
Distinct variable group:    x, A

Detailed syntax breakdown of Definition df-iord
StepHypRef Expression
1 cA . . 3  class  A
21word 4126 . 2  wff  Ord  A
31wtr 3881 . . 3  wff  Tr  A
4 vx . . . . . 6  setvar  x
54cv 1258 . . . . 5  class  x
65wtr 3881 . . . 4  wff  Tr  x
76, 4, 1wral 2323 . . 3  wff  A. x  e.  A  Tr  x
83, 7wa 101 . 2  wff  ( Tr  A  /\  A. x  e.  A  Tr  x
)
92, 8wb 102 1  wff  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
Colors of variables: wff set class
This definition is referenced by:  dford3  4131
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