Theorem tr0 5208

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 4348	. 2 ⊢ ∅ ⊆ 𝒫 ∅
2		dftr4 5202	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 231	1 ⊢ Tr ∅

Syntax hints:   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4548  Tr wtr 5196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-v 3436  df-dif 3903  df-ss 3917  df-nul 4282  df-pw 4550  df-uni 4858  df-tr 5197

This theorem is referenced by:  ord0  6356  tctr  9625  tc0  9632  r1tr  9661