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Theorem tr0 5272
Description: The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
Assertion
Ref Expression
tr0 Tr ∅

Proof of Theorem tr0
StepHypRef Expression
1 0ss 4400 . 2 ∅ ⊆ 𝒫 ∅
2 dftr4 5266 . 2 (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
31, 2mpbir 231 1 Tr ∅
Colors of variables: wff setvar class
Syntax hints:  wss 3951  c0 4333  𝒫 cpw 4600  Tr wtr 5259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-dif 3954  df-ss 3968  df-nul 4334  df-pw 4602  df-uni 4908  df-tr 5260
This theorem is referenced by:  ord0  6437  tctr  9780  tc0  9787  r1tr  9816
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