Theorem so0 4357

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 4342	. 2 |- R Po (/)
2		ral0 3548	. 2 |- A. x e. (/) A. y e. (/) A. z e. (/) ( x R y -> ( x R z \/ z R y ) )
3		df-iso 4328	. 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 944	1 |- R Or (/)

Syntax hints:   -> wi 4   \/ wo 709   A.wral 2472   (/)c0 3446   class class class wbr 4029   Po wpo 4325   Or wor 4326

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-dif 3155  df-nul 3447  df-po 4327  df-iso 4328

This theorem is referenced by: (None)