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Theorem tr0 5177
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 4316 . 2 ∅ ⊆ 𝒫 ∅
2 dftr4 5171 . 2 (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
31, 2mpbir 234 1 Tr ∅
Colors of variables: wff setvar class
Syntax hints:  wss 3871  c0 4242  𝒫 cpw 4518  Tr wtr 5166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-v 3415  df-dif 3874  df-in 3878  df-ss 3888  df-nul 4243  df-pw 4520  df-uni 4825  df-tr 5167
This theorem is referenced by:  ord0  6270  tctr  9361  tc0  9368  r1tr  9397
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