Theorem fr0 2923

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 2915	. 2 |- (R Fr (/) <-> A.x((x (_ (/) /\ x =/= (/)) -> E.y e. x (x i^i {z | zRy}) = (/)))
2		ss0 2300	. . . . 5 |- (x (_ (/) -> x = (/))
3		nne 1587	. . . . 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 987	1 |- R Fr (/)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 955  {cab 1462   =/= wne 1583  E.wrex 1644   i^i cin 2043   (_ wss 2044  (/)c0 2277   class class class wbr 2615   Fr wfr 2911

This theorem is referenced by:  we0 2940

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-fr 2913

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