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Theorem tr0 5278
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 4406 . 2 ∅ ⊆ 𝒫 ∅
2 dftr4 5272 . 2 (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
31, 2mpbir 231 1 Tr ∅
Colors of variables: wff setvar class
Syntax hints:  wss 3963  c0 4339  𝒫 cpw 4605  Tr wtr 5265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-dif 3966  df-ss 3980  df-nul 4340  df-pw 4607  df-uni 4913  df-tr 5266
This theorem is referenced by:  ord0  6439  tctr  9778  tc0  9785  r1tr  9814
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