Theorem tr0 2687

Description: The empty set is transitive.

Assertion

Ref	Expression
tr0	|- Tr (/)

Proof of Theorem tr0

Step	Hyp	Ref	Expression
1		0ss 2298	. 2 |- (/) (_ P~(/)
2		dftr4 2681	. 2 |- (Tr (/) <-> (/) (_ P~(/))
3	1, 2	mpbir 190	1 |- Tr (/)

Syntax hints:   (_ wss 2044  (/)c0 2277  P~cpw 2398  Tr wtr 2676

This theorem is referenced by:  ord0 3017  r1tr 4637

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ral 1647  df-v 1809  df-dif 2046  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-uni 2500  df-tr 2677

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