Theorem tr0 5219

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 4353	. 2 ⊢ ∅ ⊆ 𝒫 ∅
2		dftr4 5212	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 233	1 ⊢ Tr ∅

Syntax hints:   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4554  Tr wtr 5206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-v 3455  df-dif 3907  df-ss 3921  df-nul 4286  df-pw 4556  df-uni 4865  df-tr 5207

This theorem is referenced by:  ord0  6396  tctr  9690  tc0  9697  r1tr  9731  ttc0  36831