Theorem po0 2855

Description: Any relation is a partial ordering of the empty set.

Assertion

Ref	Expression
po0	|- R Po (/)

Proof of Theorem po0

Step	Hyp	Ref	Expression
1		df-po 2846	. 2 |- (R Po (/) <-> A.x e. (/) A.y e. (/) A.z e. (/) (-. xRx /\ ((xRy /\ yRz) -> xRz)))
2		noel 2287	. . 3 |- -. x e. (/)
3	2	pm2.21i 77	. 2 |- (x e. (/) -> A.y e. (/) A.z e. (/) (-. xRx /\ ((xRy /\ yRz) -> xRz)))
4	1, 3	mprgbir 1704	1 |- R Po (/)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   e. wcel 960  A.wral 1648  (/)c0 2283   class class class wbr 2624   Po wpo 2844

This theorem is referenced by:  so0 2871

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-dif 2052  df-nul 2284  df-po 2846

Copyright terms: Public domain