Theorem po0 4432

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 3611	. 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 4417	. 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 2520   (/)c0 3508   class class class wbr 4109   Po wpo 4415

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2815  df-dif 3213  df-nul 3509  df-po 4417

This theorem is referenced by:  so0  4447