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Theorem tr0 5179
 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 4353 . 2 ∅ ⊆ 𝒫 ∅
2 dftr4 5173 . 2 (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
31, 2mpbir 232 1 Tr ∅
 Colors of variables: wff setvar class Syntax hints:   ⊆ wss 3939  ∅c0 4294  𝒫 cpw 4541  Tr wtr 5168 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-v 3501  df-dif 3942  df-in 3946  df-ss 3955  df-nul 4295  df-pw 4543  df-uni 4837  df-tr 5169 This theorem is referenced by:  ord0  6240  tctr  9174  tc0  9181  r1tr  9197
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