Theorem fr0 2933

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 2925	. 2 |- (R Fr (/) <-> A.x((x (_ (/) /\ x =/= (/)) -> E.y e. x (x i^i {z | zRy}) = (/)))
2		ss0 2307	. . . . 5 |- (x (_ (/) -> x = (/))
3		nne 1592	. . . . 5 |- (-. x =/= (/) <-> x = (/))
4	2, 3	sylibr 200	. . . 4 |- (x (_ (/) -> -. x =/= (/))
5		imnan 242	. . . 4 |- ((x (_ (/) -> -. x =/= (/)) <-> -. (x (_ (/) /\ x =/= (/)))
6	4, 5	mpbi 189	. . 3 |- -. (x (_ (/) /\ x =/= (/))
7	6	pm2.21i 77	. 2 |- ((x (_ (/) /\ x =/= (/)) -> E.y e. x (x i^i {z | zRy}) = (/))
8	1, 7	mpgbir 990	1 |- R Fr (/)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 958  {cab 1466   =/= wne 1588  E.wrex 1649   i^i cin 2049   (_ wss 2050  (/)c0 2283   class class class wbr 2624   Fr wfr 2921

This theorem is referenced by:  we0 2950

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-fr 2923

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