Theorem so0 5068

Description: Any relation is a strict ordering of the empty set. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
so0	|- R Or (/)

Proof of Theorem so0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		po0 5050	. 2 |- R Po (/)
2		ral0 4076	. 2 |- A. x e. (/) A. y e. (/) ( x R y \/ x = y \/ y R x )
3		df-so 5036	. 2 |- ( R Or (/) <-> ( R Po (/) /\ A. x e. (/) A. y e. (/) ( x R y \/ x = y \/ y R x ) ) )
4	1, 2, 3	mpbir2an 955	1 |- R Or (/)

Syntax hints:   \/ w3o 1036   A.wral 2912   (/)c0 3915   class class class wbr 4653   Po wpo 5033   Or wor 5034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916  df-po 5035  df-so 5036

This theorem is referenced by:  we0  5109  wemapso2  8458