Theorem fr0 4818

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 x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffr2 4804	. 2 |- ( R Fr (/) <-> A. x ( ( x C_ (/) /\ x =/= (/) ) -> E. y e. x { z e. x | z R y } = (/) ) )
2		ss0 3768	. . . . 5 |- ( x C_ (/) -> x = (/) )
3	2	a1d 25	. . . 4 |- ( x C_ (/) -> ( -. E. y e. x { z e. x | z R y } = (/) -> x = (/) ) )
4	3	necon1ad 2660	. . 3 |- ( x C_ (/) -> ( x =/= (/) -> E. y e. x { z e. x | z R y } = (/) ) )
5	4	imp 436	. 2 |- ( ( x C_ (/) /\ x =/= (/) ) -> E. y e. x { z e. x | z R y } = (/) )
6	1, 5	mpgbir 1681	1 |- R Fr (/)

Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   = wceq 1452   =/= wne 2641   E.wrex 2757   {crab 2760   C_ wss 3390   (/)c0 3722   class class class wbr 4395   Fr wfr 4795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-fr 4798

This theorem is referenced by:  we0  4834  frsn  4910  frfi  7834  ifr0  36873