Definition df-ord 5764

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 5760	. 2 wff Ord 𝐴
3	1	wtr 4785	. . 3 wff Tr 𝐴
4		cep 5057	. . . 4 class E
5	1, 4	wwe 5101	. . 3 wff E We 𝐴
6	3, 5	wa 383	. 2 wff (Tr 𝐴 ∧ E We 𝐴)
7	2, 6	wb 196	1 wff (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))

This definition is referenced by:  ordeq  5768  ordwe  5774  ordtr  5775  trssord  5778  ordelord  5783  ord0  5815  ordon  7024  dfrecs3  7514  dford2  8555  smobeth  9446  gruina  9678  dford5  31734  dford5reg  31811  dfon2  31821