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Mirrors > Home > MPE Home > Th. List > df-ord | Structured version Visualization version GIF version |
Description: Define the ordinal predicate, which is true for a class that is transitive and is well-ordered by the epsilon relation. Variant of definition of [BellMachover] p. 468. (Contributed by NM, 17-Sep-1993.) |
Ref | Expression |
---|---|
df-ord | ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | 1 | word 5639 | . 2 wff Ord 𝐴 |
3 | 1 | wtr 4680 | . . 3 wff Tr 𝐴 |
4 | cep 4947 | . . . 4 class E | |
5 | 1, 4 | wwe 4996 | . . 3 wff E We 𝐴 |
6 | 3, 5 | wa 383 | . 2 wff (Tr 𝐴 ∧ E We 𝐴) |
7 | 2, 6 | wb 195 | 1 wff (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) |
Colors of variables: wff setvar class |
This definition is referenced by: ordeq 5647 ordwe 5653 ordtr 5654 trssord 5657 ordelord 5662 ord0 5694 ordon 6874 dfrecs3 7356 dford2 8400 smobeth 9287 gruina 9519 dford5reg 30931 dfon2 30941 |
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