Definition df-ord 6335

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 6331	. 2 wff Ord 𝐴
3	1	wtr 5214	. . 3 wff Tr 𝐴
4		cep 5537	. . . 4 class E
5	1, 4	wwe 5590	. . 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  6339  ordwe  6345  ordtr  6346  trssord  6349  ordelord  6354  ord0  6386  ordon  7753  dford5  7760  dfrecs3  8341  dford2  9573  smobeth  10539  gruina  10771  dford5reg  35770  dfon2  35780  oaun3lem1  43363