Theorem tr0 5198

Description: The empty set is transitive. (Contributed by NM, 16-Sep-1993.)

Assertion

Ref	Expression
tr0	⊢ Tr ∅

Proof of Theorem tr0

Step	Hyp	Ref	Expression
1		0ss 4327	. 2 ⊢ ∅ ⊆ 𝒫 ∅
2		dftr4 5192	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 230	1 ⊢ Tr ∅

Syntax hints:   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  Tr wtr 5187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4254  df-pw 4532  df-uni 4837  df-tr 5188

This theorem is referenced by:  ord0  6303  tctr  9429  tc0  9436  r1tr  9465