Theorem tr0 2681

Description: The empty set is transitive.

Assertion

Ref	Expression
tr0	|- Tr (/)

Proof of Theorem tr0

Step	Hyp	Ref	Expression
1		0ss 2291	. 2 |- (/) (_ P~(/)
2		dftr4 2675	. 2 |- (Tr (/) <-> (/) (_ P~(/))
3	1, 2	mpbir 190	1 |- Tr (/)

Syntax hints:   (_ wss 2037  (/)c0 2270  P~cpw 2391  Tr wtr 2670

This theorem is referenced by:  ord0 3011  r1tr 4626

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-v 1803  df-dif 2039  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-uni 2494  df-tr 2671

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