Theorem fr0 4311

Description: Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)

Assertion

Ref	Expression
fr0	|- R Fr (/)

Proof of Theorem fr0

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

Step	Hyp	Ref	Expression
1		df-frind 4292	. 2 |- ( R Fr (/) <-> A. sFrFor R (/) s )
2		0ss 3432	. . . 4 |- (/) C_ s
3	2	a1i 9	. . 3 |- ( A. x e. (/) ( A. y e. (/) ( y R x -> y e. s ) -> x e. s ) -> (/) C_ s )
4		df-frfor 4291	. . 3 |- (FrFor R (/) s <-> ( A. x e. (/) ( A. y e. (/) ( y R x -> y e. s ) -> x e. s ) -> (/) C_ s ) )
5	3, 4	mpbir 145	. 2 |- FrFor R (/) s
6	1, 5	mpgbir 1433	1 |- R Fr (/)

Syntax hints:   -> wi 4   A.wral 2435   C_ wss 3102   (/)c0 3394   class class class wbr 3965  FrFor wfrfor 4287   Fr wfr 4288

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395  df-frfor 4291  df-frind 4292

This theorem is referenced by:  we0  4321