Theorem fr0 2927

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 2919	. 2 |- (R Fr (/) <-> A.x((x (_ (/) /\ x =/= (/)) -> E.y e. x (x i^i {z | zRy}) = (/)))
2		ss0 2303	. . . . 5 |- (x (_ (/) -> x = (/))
3		nne 1589	. . . . 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 988	1 |- R Fr (/)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 956  {cab 1463   =/= wne 1585  E.wrex 1646   i^i cin 2046   (_ wss 2047  (/)c0 2280   class class class wbr 2619   Fr wfr 2915

This theorem is referenced by:  we0 2944

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-fr 2917

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