Theorem so0 4416

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 4401	. 2 |- R Po (/)
2		ral0 3593	. 2 |- A. x e. (/) A. y e. (/) A. z e. (/) ( x R y -> ( x R z \/ z R y ) )
3		df-iso 4387	. 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 948	1 |- R Or (/)

Syntax hints:   -> wi 4   \/ wo 713   A.wral 2508   (/)c0 3491   class class class wbr 4082   Po wpo 4384   Or wor 4385

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-dif 3199  df-nul 3492  df-po 4386  df-iso 4387

This theorem is referenced by: (None)