Theorem tr0 4956

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 4168	. 2 ⊢ ∅ ⊆ 𝒫 ∅
2		dftr4 4950	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 223	1 ⊢ Tr ∅

Syntax hints:   ⊆ wss 3769  ∅c0 4115  𝒫 cpw 4349  Tr wtr 4945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-v 3387  df-dif 3772  df-in 3776  df-ss 3783  df-nul 4116  df-pw 4351  df-uni 4629  df-tr 4946

This theorem is referenced by:  ord0  5993  tctr  8866  tc0  8873  r1tr  8889