Theorem tr0 5232

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 4358	. 2 ⊢ ∅ ⊆ 𝒫 ∅
2		dftr4 5225	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 234	1 ⊢ Tr ∅

Syntax hints:   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4563  Tr wtr 5219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-v 3457  df-dif 3909  df-ss 3923  df-nul 4288  df-pw 4565  df-uni 4874  df-tr 5220

This theorem is referenced by:  ord0  6417  tctr  9708  tc0  9715  r1tr  9749  ttc0  36996