Theorem tr0 5224

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 4349	. 2 ⊢ ∅ ⊆ 𝒫 ∅
2		dftr4 5217	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 234	1 ⊢ Tr ∅

This proof depends on syntax axioms:   ⊆ wss 3895  ∅c0 4278  𝒫 cpw 4556  Tr wtr 5211

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-ss 3896  df-dif 4047  df-nul 4279  df-pw 4558  df-uni 4867  df-tr 5212

This theorem is used by:  ord0  6417  tctr  9743  tc0  9750  r1tr  9787  ttc0  37388