Theorem fr0 4382

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 4363	. 2 |- ( R Fr (/) <-> A. sFrFor R (/) s )
2		0ss 3485	. . . 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 4362	. . 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 146	. 2 |- FrFor R (/) s
6	1, 5	mpgbir 1464	1 |- R Fr (/)

Syntax hints:   -> wi 4   A.wral 2472   C_ wss 3153   (/)c0 3446   class class class wbr 4029  FrFor wfrfor 4358   Fr wfr 4359

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3447  df-frfor 4362  df-frind 4363

This theorem is referenced by:  we0  4392