Theorem tr0 4797

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 4005	. 2 ⊢ ∅ ⊆ 𝒫 ∅
2		dftr4 4790	. 2 ⊢ (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
3	1, 2	mpbir 221	1 ⊢ Tr ∅

Syntax hints:   ⊆ wss 3607  ∅c0 3948  𝒫 cpw 4191  Tr wtr 4785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-uni 4469  df-tr 4786

This theorem is referenced by:  ord0  5815  tctr  8654  tc0  8661  r1tr  8677