Theorem tr0 5268

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 4388	. 2 ⊢ ∅ ⊆ 𝒫 ∅
2		dftr4 5262	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 230	1 ⊢ Tr ∅

Syntax hints:   ⊆ wss 3940  ∅c0 4314  𝒫 cpw 4594  Tr wtr 5255

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 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-v 3468  df-dif 3943  df-in 3947  df-ss 3957  df-nul 4315  df-pw 4596  df-uni 4900  df-tr 5256

This theorem is referenced by:  ord0  6407  tctr  9731  tc0  9738  r1tr  9767