![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5760 | . 2 wff Ord 𝐴 |
3 | 1 | wtr 4785 | . . 3 wff Tr 𝐴 |
4 | cep 5057 | . . . 4 class E | |
5 | 1, 4 | wwe 5101 | . . 3 wff E We 𝐴 |
6 | 3, 5 | wa 383 | . 2 wff (Tr 𝐴 ∧ E We 𝐴) |
7 | 2, 6 | wb 196 | 1 wff (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) |
Colors of variables: wff setvar class |
This definition is referenced by: ordeq 5768 ordwe 5774 ordtr 5775 trssord 5778 ordelord 5783 ord0 5815 ordon 7024 dfrecs3 7514 dford2 8555 smobeth 9446 gruina 9678 dford5 31734 dford5reg 31811 dfon2 31821 |
Copyright terms: Public domain | W3C validator |