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 4357	. 2 ⊢ ∅ ⊆ 𝒫 ∅
2		dftr4 5218	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 234	1 ⊢ Tr ∅

Syntax hints:   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558  Tr wtr 5212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-v 3459  df-dif 3910  df-ss 3924  df-nul 4289  df-pw 4560  df-uni 4869  df-tr 5213

This theorem is referenced by:  ord0  6404  tctr  9695  tc0  9702  r1tr  9736  ttc0  36880