Theorem we0 4346

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 4336	. 2 |- R Fr (/)
2		ral0 3516	. 2 |- A. x e. (/) A. y e. (/) A. z e. (/) ( ( x R y /\ y R z ) -> x R z )
3		df-wetr 4319	. 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 937	1 |- R We (/)

Syntax hints:   -> wi 4   /\ wa 103   A.wral 2448   (/)c0 3414   class class class wbr 3989   Fr wfr 4313   We wwe 4315

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-frfor 4316  df-frind 4317  df-wetr 4319

This theorem is referenced by: (None)