Theorem we0 4464

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 4454	. 2 |- R Fr (/)
2		ral0 3598	. 2 |- A. x e. (/) A. y e. (/) A. z e. (/) ( ( x R y /\ y R z ) -> x R z )
3		df-wetr 4437	. 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 951	1 |- R We (/)

Syntax hints:   -> wi 4   /\ wa 104   A.wral 2511   (/)c0 3496   class class class wbr 4093   Fr wfr 4431   We wwe 4433

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-nul 3497  df-frfor 4434  df-frind 4435  df-wetr 4437

This theorem is referenced by: (None)