Theorem fr0 4847

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 4833	. 2 |- ( R Fr (/) <-> A. x ( ( x C_ (/) /\ x =/= (/) ) -> E. y e. x { z e. x | z R y } = (/) ) )
2		ss0 3815	. . . . 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 2670	. . 3 |- ( x C_ (/) -> ( x =/= (/) -> E. y e. x { z e. x | z R y } = (/) ) )
5	4	imp 427	. 2 |- ( ( x C_ (/) /\ x =/= (/) ) -> E. y e. x { z e. x | z R y } = (/) )
6	1, 5	mpgbir 1627	1 |- R Fr (/)

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   = wceq 1398   =/= wne 2649   E.wrex 2805   {crab 2808   C_ wss 3461   (/)c0 3783   class class class wbr 4439   Fr wfr 4824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784  df-fr 4827

This theorem is referenced by:  we0  4863  frsn  5059  frfi  7757  ifr0  31603