Definition df-ord 6365

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 6361	. 2 wff Ord 𝐴
3	1	wtr 5265	. . 3 wff Tr 𝐴
4		cep 5579	. . . 4 class E
5	1, 4	wwe 5630	. . 3 wff E We 𝐴
6	3, 5	wa 397	. 2 wff (Tr 𝐴 ∧ E We 𝐴)
7	2, 6	wb 205	1 wff (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))

This definition is referenced by:  ordeq  6369  ordwe  6375  ordtr  6376  trssord  6379  ordelord  6384  ord0  6415  ordon  7761  dford5  7768  dfrecs3  8369  dfrecs3OLD  8370  dford2  9612  smobeth  10578  gruina  10810  dford5reg  34743  dfon2  34753  oaun3lem1  42110