Theorem fr0 4329

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 4310	. 2 |- ( R Fr (/) <-> A. sFrFor R (/) s )
2		0ss 3447	. . . 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 4309	. . 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 1441	1 |- R Fr (/)

Syntax hints:   -> wi 4   A.wral 2444   C_ wss 3116   (/)c0 3409   class class class wbr 3982  FrFor wfrfor 4305   Fr wfr 4306

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-frfor 4309  df-frind 4310

This theorem is referenced by:  we0  4339