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Theorem tr0 5239
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 4360 . 2 ∅ ⊆ 𝒫 ∅
2 dftr4 5233 . 2 (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
31, 2mpbir 230 1 Tr ∅
Colors of variables: wff setvar class
Syntax hints:  wss 3914  c0 4286  𝒫 cpw 4564  Tr wtr 5226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-v 3449  df-dif 3917  df-in 3921  df-ss 3931  df-nul 4287  df-pw 4566  df-uni 4870  df-tr 5227
This theorem is referenced by:  ord0  6374  tctr  9684  tc0  9691  r1tr  9720
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