Theorem tr0 5231

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 4353	. 2 ⊢ ∅ ⊆ 𝒫 ∅
2		dftr4 5224	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 234	1 ⊢ Tr ∅

This proof depends on syntax axioms:   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560  Tr wtr 5218

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-v 3456  df-ss 3901  df-dif 4050  df-nul 4282  df-pw 4562  df-uni 4873  df-tr 5219

This theorem is used by:  ord0  6443  tctr  9776  tc0  9783  r1tr  9820  ttc0  37510