Theorem fr0 4273

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 4254	. 2 |- ( R Fr (/) <-> A. sFrFor R (/) s )
2		0ss 3401	. . . 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 4253	. . 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 1429	1 |- R Fr (/)

Syntax hints:   -> wi 4   A.wral 2416   C_ wss 3071   (/)c0 3363   class class class wbr 3929  FrFor wfrfor 4249   Fr wfr 4250

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-frfor 4253  df-frind 4254

This theorem is referenced by:  we0  4283