Theorem po0 2840

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 2831	. 2 |- (R Po (/) <-> A.x e. (/) A.y e. (/) A.z e. (/) (-. xRx /\ ((xRy /\ yRz) -> xRz)))
2		noel 2274	. . 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 1693	1 |- R Po (/)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   e. wcel 955  A.wral 1637  (/)c0 2270   class class class wbr 2609   Po wpo 2829

This theorem is referenced by:  so0 2856

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-v 1803  df-dif 2039  df-nul 2271  df-po 2831

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