Theorem tr0 5236

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 4360	. 2 ⊢ ∅ ⊆ 𝒫 ∅
2		dftr4 5229	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 234	1 ⊢ Tr ∅

This proof depends on syntax axioms:   ⊆ wss 3908  ∅c0 4289  𝒫 cpw 4567  Tr wtr 5223

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-v 3460  df-dif 3911  df-ss 3925  df-nul 4290  df-pw 4569  df-uni 4878  df-tr 5224

This theorem is used by:  ord0  6422  tctr  9717  tc0  9724  r1tr  9758  ttc0  37059