Definition df-ord 6378

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.)

Assertion

Ref	Expression
df-ord	⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))

Detailed syntax breakdown of Definition df-ord

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	word 6374	. 2 wff Ord 𝐴
3	1	wtr 5263	. . 3 wff Tr 𝐴
4		cep 5577	. . . 4 class E
5	1, 4	wwe 5629	. . 3 wff E We 𝐴
6	3, 5	wa 395	. 2 wff (Tr 𝐴 ∧ E We 𝐴)
7	2, 6	wb 206	1 wff (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))

This definition is referenced by:  ordeq  6382  ordwe  6388  ordtr  6389  trssord  6392  ordelord  6397  ord0  6428  ordon  7783  dford5  7790  dfrecs3  8398  dfrecs3OLD  8399  dford2  9644  smobeth  10610  gruina  10842  dford5reg  35719  dfon2  35729  oaun3lem1  43323