Theorem po0 4376

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 3570	. 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 4361	. 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 2486   (/)c0 3468   class class class wbr 4059   Po wpo 4359

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-dif 3176  df-nul 3469  df-po 4361

This theorem is referenced by:  so0  4391