Theorem tr0 5227

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 4363	. 2 ⊢ ∅ ⊆ 𝒫 ∅
2		dftr4 5221	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 231	1 ⊢ Tr ∅

Syntax hints:   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  Tr wtr 5214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-v 3449  df-dif 3917  df-ss 3931  df-nul 4297  df-pw 4565  df-uni 4872  df-tr 5215

This theorem is referenced by:  ord0  6386  tctr  9693  tc0  9700  r1tr  9729