Theorem tr0 5296

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 4423	. 2 ⊢ ∅ ⊆ 𝒫 ∅
2		dftr4 5290	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 231	1 ⊢ Tr ∅

Syntax hints:   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  Tr wtr 5283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-v 3490  df-dif 3979  df-ss 3993  df-nul 4353  df-pw 4624  df-uni 4932  df-tr 5284

This theorem is referenced by:  ord0  6448  tctr  9809  tc0  9816  r1tr  9845