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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. (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 6184 | . 2 wff Ord 𝐴 |
3 | 1 | wtr 5164 | . . 3 wff Tr 𝐴 |
4 | cep 5458 | . . . 4 class E | |
5 | 1, 4 | wwe 5507 | . . 3 wff E We 𝐴 |
6 | 3, 5 | wa 396 | . 2 wff (Tr 𝐴 ∧ E We 𝐴) |
7 | 2, 6 | wb 207 | 1 wff (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) |
Colors of variables: wff setvar class |
This definition is referenced by: ordeq 6192 ordwe 6198 ordtr 6199 trssord 6202 ordelord 6207 ord0 6237 ordon 7486 dfrecs3 8000 dford2 9072 smobeth 9997 gruina 10229 dford5 32855 dford5reg 32925 dfon2 32935 |
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