Theorem we0 4360

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 4350	. 2 |- R Fr (/)
2		ral0 3524	. 2 |- A. x e. (/) A. y e. (/) A. z e. (/) ( ( x R y /\ y R z ) -> x R z )
3		df-wetr 4333	. 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 942	1 |- R We (/)

Syntax hints:   -> wi 4   /\ wa 104   A.wral 2455   (/)c0 3422   class class class wbr 4002   Fr wfr 4327   We wwe 4329

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423  df-frfor 4330  df-frind 4331  df-wetr 4333

This theorem is referenced by: (None)