Definition df-ord 5628

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 5624	. 2 wff Ord 𝐴
3	1	wtr 4674	. . 3 wff Tr 𝐴
4		cep 4936	. . . 4 class E
5	1, 4	wwe 4985	. . 3 wff E We 𝐴
6	3, 5	wa 382	. 2 wff (Tr 𝐴 ∧ E We 𝐴)
7	2, 6	wb 194	1 wff (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))

This definition is referenced by:  ordeq  5632  ordwe  5638  ordtr  5639  trssord  5642  ordelord  5647  ord0  5679  ordon  6851  dfrecs3  7333  dford2  8377  smobeth  9264  gruina  9496  dford5reg  30724  dfon2  30734