Theorem fr0 2179

Description: Any relation is founded on the empty set.

Assertion

Ref	Expression
fr0	|- R Fr (/)

Proof of Theorem fr0

Step	Hyp	Ref	Expression
1		dffr2 2171	. 2 |- (R Fr (/) <-> A.x((x (_ (/) /\ -. x = (/)) -> E.y e. x (x i^i {z | zRy}) = (/)))
2		ss0 1727	. . . 4 |- (x (_ (/) -> x = (/))
3		iman 205	. . . 4 |- ((x (_ (/) -> x = (/)) <-> -. (x (_ (/) /\ -. x = (/)))
4	2, 3	mpbi 164	. . 3 |- -. (x (_ (/) /\ -. x = (/))
5	4	pm2.21i 73	. 2 |- ((x (_ (/) /\ -. x = (/)) -> E.y e. x (x i^i {z | zRy}) = (/))
6	1, 5	mpgbir 686	1 |- R Fr (/)

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  {cab 1090   = wceq 1091  E.wrex 1202   i^i cin 1486   (_ wss 1487  (/)c0 1707   class class class wbr 2054   Fr wfr 2061

This theorem is referenced by:  we0 2196

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-fr 2169

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