Definition df-tr 5192

Description: Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 6017). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 5193 (which is suggestive of the word "transitive"), dftr3 5195, dftr4 5196, dftr5 5194, and (when 𝐴 is a set) unisuc 6342. 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 5191	. 2 wff Tr 𝐴
3	1	cuni 4839	. . 3 class ∪ 𝐴
4	3, 1	wss 3887	. 2 wff ∪ 𝐴 ⊆ 𝐴
5	2, 4	wb 205	1 wff (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)

This definition is referenced by:  dftr2  5193  dftr4  5196  treq  5197  trv  5203  pwtr  5368  unisuc  6342  orduniss  6360  onuninsuci  7687  trcl  9486  tc2  9500  r1tr2  9535  tskuni  10539  untangtr  33655  hfuni  34486  nlimsuc  41048