Theorem tr0 5225

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 4350	. 2 ⊢ ∅ ⊆ 𝒫 ∅
2		dftr4 5218	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 234	1 ⊢ Tr ∅

This proof depends on syntax axioms:   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  Tr wtr 5212

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 2147  ax-9 2155  ax-ext 2732

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 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-dif 3902  df-ss 3916  df-nul 4280  df-pw 4559  df-uni 4868  df-tr 5213

This theorem is used by:  ord0  6412  tctr  9717  tc0  9724  r1tr  9758  ttc0  37126