Theorem po0 4359

Description: Any relation is a partial ordering of the empty set. (Contributed by NM, 28-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
po0	|- R Po (/)

Proof of Theorem po0

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 3562	. 2 |- A. x e. (/) A. y e. (/) A. z e. (/) ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) )
2		df-po 4344	. 2 |- ( R Po (/) <-> A. x e. (/) A. y e. (/) A. z e. (/) ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) )
3	1, 2	mpbir 146	1 |- R Po (/)

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wral 2484   (/)c0 3460   class class class wbr 4045   Po wpo 4342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-dif 3168  df-nul 3461  df-po 4344

This theorem is referenced by:  so0  4374