Definition df-ord 6069

Description: Define the ordinal predicate, which is true for a class that is transitive and is well-ordered by the epsilon relation. Variant of definition of [BellMachover] p. 468. (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 6065	. 2 wff Ord 𝐴
3	1	wtr 5063	. . 3 wff Tr 𝐴
4		cep 5352	. . . 4 class E
5	1, 4	wwe 5401	. . 3 wff E We 𝐴
6	3, 5	wa 396	. 2 wff (Tr 𝐴 ∧ E We 𝐴)
7	2, 6	wb 207	1 wff (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))

This definition is referenced by:  ordeq  6073  ordwe  6079  ordtr  6080  trssord  6083  ordelord  6088  ord0  6118  ordon  7354  dfrecs3  7861  dford2  8929  smobeth  9854  gruina  10086  dford5  32565  dford5reg  32635  dfon2  32645