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Theorem tr0 5279
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 4398 . 2 ∅ ⊆ 𝒫 ∅
2 dftr4 5273 . 2 (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
31, 2mpbir 230 1 Tr ∅
Colors of variables: wff setvar class
Syntax hints:  wss 3944  c0 4322  𝒫 cpw 4604  Tr wtr 5266
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-v 3463  df-dif 3947  df-ss 3961  df-nul 4323  df-pw 4606  df-uni 4910  df-tr 5267
This theorem is referenced by:  ord0  6424  tctr  9765  tc0  9772  r1tr  9801
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