Theorem fr0 4399

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 4380	. 2 |- ( R Fr (/) <-> A. sFrFor R (/) s )
2		0ss 3499	. . . 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 4379	. . 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 1476	1 |- R Fr (/)

Syntax hints:   -> wi 4   A.wral 2484   C_ wss 3166   (/)c0 3460   class class class wbr 4045  FrFor wfrfor 4375   Fr wfr 4376

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3461  df-frfor 4379  df-frind 4380

This theorem is referenced by:  we0  4409