Definition df-ord 6315

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 6311	. 2 wff Ord 𝐴
3	1	wtr 5181	. . 3 wff Tr 𝐴
4		cep 5519	. . . 4 class E
5	1, 4	wwe 5572	. . 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  6319  ordwe  6325  ordtr  6326  trssord  6329  ordelord  6334  ord0  6366  ordon  7720  dford5  7727  dfrecs3  8301  dford2  9530  smobeth  10498  gruina  10730  dford5reg  35950  dfon2  35960  oaun3lem1  43790