Theorem tr0 5272

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 4392	. 2 ⊢ ∅ ⊆ 𝒫 ∅
2		dftr4 5266	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 230	1 ⊢ Tr ∅

Syntax hints:   ⊆ wss 3944  ∅c0 4318  𝒫 cpw 4598  Tr wtr 5259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-v 3471  df-dif 3947  df-in 3951  df-ss 3961  df-nul 4319  df-pw 4600  df-uni 4904  df-tr 5260

This theorem is referenced by:  ord0  6416  tctr  9755  tc0  9762  r1tr  9791