Theorem fr0 5093

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 5079	. 2 |- ( R Fr (/) <-> A. x ( ( x C_ (/) /\ x =/= (/) ) -> E. y e. x { z e. x | z R y } = (/) ) )
2		ss0 3974	. . . . 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 2811	. . 3 |- ( x C_ (/) -> ( x =/= (/) -> E. y e. x { z e. x | z R y } = (/) ) )
5	4	imp 445	. 2 |- ( ( x C_ (/) /\ x =/= (/) ) -> E. y e. x { z e. x | z R y } = (/) )
6	1, 5	mpgbir 1726	1 |- R Fr (/)

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   =/= wne 2794   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Fr wfr 5070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-fr 5073

This theorem is referenced by:  we0  5109  frsn  5189  frfi  8205  ifr0  38654