Theorem fr0 3636

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 3627	. 2 |- (R Fr (/) <-> A.x((x C_ (/) /\ x =/= (/)) -> E.y e. x (x i^i {z | zRy}) = (/)))
2		ss0 2902	. . . . 5 |- (x C_ (/) -> x = (/))
3		nne 2021	. . . . 5 |- (-. x =/= (/) <-> x = (/))
4	2, 3	sylibr 217	. . . 4 |- (x C_ (/) -> -. x =/= (/))
5		imnan 261	. . . 4 |- ((x C_ (/) -> -. x =/= (/)) <-> -. (x C_ (/) /\ x =/= (/)))
6	4, 5	mpbi 206	. . 3 |- -. (x C_ (/) /\ x =/= (/))
7	6	pm2.21i 93	. 2 |- ((x C_ (/) /\ x =/= (/)) -> E.y e. x (x i^i {z | zRy}) = (/))
8	1, 7	mpgbir 1334	1 |- R Fr (/)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298  {cab 1871   =/= wne 2017  E.wrex 2106   i^i cin 2592   C_ wss 2593  (/)c0 2875   class class class wbr 3338   Fr wfr 3623

This theorem is referenced by:  we0 3653  frfi 15771  ifr0 16427

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-fr 3625

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