Theorem we0 4339

Description: Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)

Assertion

Ref	Expression
we0	|- R We (/)

Proof of Theorem we0

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

Step	Hyp	Ref	Expression
1		fr0 4329	. 2 |- R Fr (/)
2		ral0 3510	. 2 |- A. x e. (/) A. y e. (/) A. z e. (/) ( ( x R y /\ y R z ) -> x R z )
3		df-wetr 4312	. 2 |- ( R We (/) <-> ( R Fr (/) /\ A. x e. (/) A. y e. (/) A. z e. (/) ( ( x R y /\ y R z ) -> x R z ) ) )
4	1, 2, 3	mpbir2an 932	1 |- R We (/)

Syntax hints:   -> wi 4   /\ wa 103   A.wral 2444   (/)c0 3409   class class class wbr 3982   Fr wfr 4306   We wwe 4308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-frfor 4309  df-frind 4310  df-wetr 4312

This theorem is referenced by: (None)