Theorem po0 4769

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 3873	. 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 4754	. 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 213	1 |- R Po (/)

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   A.wral 2736   (/)c0 3730   class class class wbr 4401   Po wpo 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ral 2741  df-v 3046  df-dif 3406  df-nul 3731  df-po 4754

This theorem is referenced by:  so0  4787  posn  4902  dfpo2  30388  ipo0  36796