Definition df-tr 5265

Description: Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 6132). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 5266 (which is suggestive of the word "transitive"), dftr2c 5267, dftr3 5270, dftr4 5271, dftr5 5268, and (when 𝐴 is a set) unisuc 6464. 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 5264	. 2 wff Tr 𝐴
3	1	cuni 4911	. . 3 class ∪ 𝐴
4	3, 1	wss 3962	. 2 wff ∪ 𝐴 ⊆ 𝐴
5	2, 4	wb 206	1 wff (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)

This definition is referenced by:  dftr2  5266  dftr4  5271  treq  5272  trv  5278  pwtr  5462  unisucg  6463  orduniss  6482  onuninsuci  7860  trcl  9765  tc2  9779  r1tr2  9814  tskuni  10820  untangtr  35693  hfuni  36165