Theorem so0 2865

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 2850	. . 3 |- (R Or (/) <-> (R Po (/) /\ A.x e. (/) A.y e. (/) (xRy \/ x = y \/ yRx)))
2		po0 2849	. . 3 |- R Po (/)
3	1, 2	mpbiran 728	. 2 |- (R Or (/) <-> A.x e. (/) A.y e. (/) (xRy \/ x = y \/ yRx))
4		noel 2284	. . 3 |- -. x e. (/)
5	4	pm2.21i 77	. 2 |- (x e. (/) -> A.y e. (/) (xRy \/ x = y \/ yRx))
6	3, 5	mprgbir 1701	1 |- R Or (/)

Syntax hints:   \/ w3o 774   = wceq 956   e. wcel 958  A.wral 1645  (/)c0 2280   class class class wbr 2619   Po wpo 2838   Or wor 2839

This theorem is referenced by:  we0 2944

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-dif 2049  df-nul 2281  df-po 2840  df-so 2850

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