Theorem tr0 5210

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 5204	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 231	1 ⊢ Tr ∅

Syntax hints:   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4550  Tr wtr 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-v 3438  df-dif 3905  df-ss 3919  df-nul 4284  df-pw 4552  df-uni 4860  df-tr 5199

This theorem is referenced by:  ord0  6360  tctr  9630  tc0  9637  r1tr  9666