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Mirrors > Home > ILE Home > Th. List > df-iord | GIF version |
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 4070 instead for naming consistency with set.mm. (New usage is discouraged.) |
Ref | Expression |
---|---|
df-iord | ⊢ (Ord A ↔ (Tr A ∧ ∀x ∈ A Tr x)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class A | |
2 | 1 | word 4065 | . 2 wff Ord A |
3 | 1 | wtr 3845 | . . 3 wff Tr A |
4 | vx | . . . . . 6 setvar x | |
5 | 4 | cv 1241 | . . . . 5 class x |
6 | 5 | wtr 3845 | . . . 4 wff Tr x |
7 | 6, 4, 1 | wral 2300 | . . 3 wff ∀x ∈ A Tr x |
8 | 3, 7 | wa 97 | . 2 wff (Tr A ∧ ∀x ∈ A Tr x) |
9 | 2, 8 | wb 98 | 1 wff (Ord A ↔ (Tr A ∧ ∀x ∈ A Tr x)) |
Colors of variables: wff set class |
This definition is referenced by: dford3 4070 |
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