Theorem so0 2856

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

Assertion

Ref	Expression
so0	|- R Or (/)

Proof of Theorem so0

Step	Hyp	Ref	Expression
1		df-so 2841	. . 3 |- (R Or (/) <-> (R Po (/) /\ A.x e. (/) A.y e. (/) (xRy \/ x = y \/ yRx)))
2		po0 2840	. . 3 |- R Po (/)
3	1, 2	mpbiran 726	. 2 |- (R Or (/) <-> A.x e. (/) A.y e. (/) (xRy \/ x = y \/ yRx))
4		noel 2274	. . 3 |- -. x e. (/)
5	4	pm2.21i 77	. 2 |- (x e. (/) -> A.y e. (/) (xRy \/ x = y \/ yRx))
6	3, 5	mprgbir 1693	1 |- R Or (/)

Syntax hints:   \/ w3o 772   = wceq 953   e. wcel 955  A.wral 1637  (/)c0 2270   class class class wbr 2609   Po wpo 2829   Or wor 2830

This theorem is referenced by:  we0 2934

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  df-so 2841

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