Theorem tr0 5199

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 4335	. 2 ⊢ ∅ ⊆ 𝒫 ∅
2		dftr4 5192	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 232	1 ⊢ Tr ∅

Syntax hints:   ⊆ wss 3890  ∅c0 4268  𝒫 cpw 4536  Tr wtr 5186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-v 3434  df-dif 3893  df-ss 3907  df-nul 4269  df-pw 4538  df-uni 4846  df-tr 5187

This theorem is referenced by:  ord0  6371  tctr  9657  tc0  9664  r1tr  9698  ttc0  36742