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Definition df-iord 4123
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 4124 instead for naming consistency with set.mm. (New usage is discouraged.)
Assertion
Ref Expression
df-iord (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
Distinct variable group:   𝑥,𝐴

Detailed syntax breakdown of Definition df-iord
StepHypRef Expression
1 cA . . 3 class 𝐴
21word 4119 . 2 wff Ord 𝐴
31wtr 3877 . . 3 wff Tr 𝐴
4 vx . . . . . 6 setvar 𝑥
54cv 1284 . . . . 5 class 𝑥
65wtr 3877 . . . 4 wff Tr 𝑥
76, 4, 1wral 2349 . . 3 wff 𝑥𝐴 Tr 𝑥
83, 7wa 102 . 2 wff (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥)
92, 8wb 103 1 wff (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
Colors of variables: wff set class
This definition is referenced by:  dford3  4124
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