Theorem so0 4466

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 z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		po0 4451	. 2 |- R Po (/)
2		ral0 3626	. 2 |- A. x e. (/) A. y e. (/) A. z e. (/) ( x R y -> ( x R z \/ z R y ) )
3		df-iso 4437	. 2 |- ( R Or (/) <-> ( R Po (/) /\ A. x e. (/) A. y e. (/) A. z e. (/) ( x R y -> ( x R z \/ z R y ) ) ) )
4	1, 2, 3	mpbir2an 955	1 |- R Or (/)

Syntax hints:   -> wi 4   \/ wo 720   A.wral 2528   (/)c0 3520   class class class wbr 4125   Po wpo 4434   Or wor 4435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-dif 3222  df-nul 3521  df-po 4436  df-iso 4437

This theorem is referenced by: (None)