Definition df-ord 6194

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. (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 6190	. 2 wff Ord 𝐴
3	1	wtr 5172	. . 3 wff Tr 𝐴
4		cep 5464	. . . 4 class E
5	1, 4	wwe 5513	. . 3 wff E We 𝐴
6	3, 5	wa 398	. 2 wff (Tr 𝐴 ∧ E We 𝐴)
7	2, 6	wb 208	1 wff (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))

This definition is referenced by:  ordeq  6198  ordwe  6204  ordtr  6205  trssord  6208  ordelord  6213  ord0  6243  ordon  7498  dfrecs3  8009  dford2  9083  smobeth  10008  gruina  10240  dford5  32957  dford5reg  33027  dfon2  33037