Definition df-ord 6354

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 6350	. 2 wff Ord 𝐴
3	1	wtr 5211	. . 3 wff Tr 𝐴
4		cep 5546	. . . 4 class E
5	1, 4	wwe 5599	. . 3 wff E We 𝐴
6	3, 5	wa 401	. 2 wff (Tr 𝐴 ∧ E We 𝐴)
7	2, 6	wb 209	1 wff (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))

This definition is used by:  ordeq  6358  ordwe  6364  ordtr  6365  trssord  6368  ordelord  6373  ord0  6406  ordon  7774  dford5  7781  dfrecs3  8358  dford2  9599  smobeth  10642  gruina  10874  dford5reg  36466  dfon2  36476  oaun3lem1  44319