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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 membership relation. Variant of definition of
[BellMachover] p. 468.
Some sources will define a notation for ordinal order corresponding to < and ≤ but we just use ∈ and ⊆ respectively. (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 6190 | . 2 wff Ord 𝐴 |
3 | 1 | wtr 5146 | . . 3 wff Tr 𝐴 |
4 | cep 5444 | . . . 4 class E | |
5 | 1, 4 | wwe 5493 | . . 3 wff E We 𝐴 |
6 | 3, 5 | wa 399 | . 2 wff (Tr 𝐴 ∧ E We 𝐴) |
7 | 2, 6 | wb 209 | 1 wff (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) |
Colors of variables: wff setvar class |
This definition is referenced by: ordeq 6198 ordwe 6204 ordtr 6205 trssord 6208 ordelord 6213 ord0 6243 ordon 7539 dfrecs3 8087 dford2 9213 smobeth 10165 gruina 10397 dford5 33340 dford5reg 33428 dfon2 33438 |
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