Definition df-tr 4723

Description: Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 5477). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4724 (which is suggestive of the word "transitive"), dftr3 4726, dftr4 4727, dftr5 4725, and (when 𝐴 is a set) unisuc 5770. The term "complete" is used instead of "transitive" in Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.)

Assertion

Ref	Expression
df-tr	⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)

Detailed syntax breakdown of Definition df-tr

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	wtr 4722	. 2 wff Tr 𝐴
3	1	cuni 4409	. . 3 class ∪ 𝐴
4	3, 1	wss 3560	. 2 wff ∪ 𝐴 ⊆ 𝐴
5	2, 4	wb 196	1 wff (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)

This definition is referenced by:  dftr2  4724  dftr4  4727  treq  4728  trv  4735  pwtr  4892  unisuc  5770  orduniss  5790  onuninsuci  7002  trcl  8564  tc2  8578  r1tr2  8600  tskuni  9565  untangtr  31352  hfuni  31986