Definition df-ord 6385

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 6381	. 2 wff Ord 𝐴
3	1	wtr 5257	. . 3 wff Tr 𝐴
4		cep 5581	. . . 4 class E
5	1, 4	wwe 5634	. . 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  6389  ordwe  6395  ordtr  6396  trssord  6399  ordelord  6404  ord0  6435  ordon  7793  dford5  7800  dfrecs3  8408  dfrecs3OLD  8409  dford2  9656  smobeth  10622  gruina  10854  dford5reg  35761  dfon2  35771  oaun3lem1  43365