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Mirrors > Home > MPE Home > Th. List > df-tr | Structured version Visualization version GIF version |
Description: Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 5658). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4898 (which is suggestive of the word "transitive"), dftr3 4900, dftr4 4901, dftr5 4899, and (when 𝐴 is a set) unisuc 5954. The term "complete" is used instead of "transitive" in Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.) |
Ref | Expression |
---|---|
df-tr | ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | 1 | wtr 4896 | . 2 wff Tr 𝐴 |
3 | 1 | cuni 4580 | . . 3 class ∪ 𝐴 |
4 | 3, 1 | wss 3707 | . 2 wff ∪ 𝐴 ⊆ 𝐴 |
5 | 2, 4 | wb 196 | 1 wff (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) |
Colors of variables: wff setvar class |
This definition is referenced by: dftr2 4898 dftr4 4901 treq 4902 trv 4909 pwtr 5062 unisuc 5954 orduniss 5974 onuninsuci 7197 trcl 8769 tc2 8783 r1tr2 8805 tskuni 9789 untangtr 31890 hfuni 32589 |
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