Theorem fr0 4783

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 4769	. 2 |- ( R Fr (/) <-> A. x ( ( x C_ (/) /\ x =/= (/) ) -> E. y e. x { z e. x | z R y } = (/) ) )
2		ss0 3752	. . . . 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 2661	. . 3 |- ( x C_ (/) -> ( x =/= (/) -> E. y e. x { z e. x | z R y } = (/) ) )
5	4	imp 429	. 2 |- ( ( x C_ (/) /\ x =/= (/) ) -> E. y e. x { z e. x | z R y } = (/) )
6	1, 5	mpgbir 1596	1 |- R Fr (/)

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1370   =/= wne 2641   E.wrex 2793   {crab 2796   C_ wss 3412   (/)c0 3721   class class class wbr 4376   Fr wfr 4760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-dif 3415  df-in 3419  df-ss 3426  df-nul 3722  df-fr 4763

This theorem is referenced by:  we0  4799  frsn  4993  frfi  7644  ifr0  29830