Theorem n0moeu 2887

Description: A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)

Assertion

Ref	Expression
n0moeu	|- (A =/= (/) -> (E*x x e. A <-> E!x x e. A))

Distinct variable group:   x,A

Proof of Theorem n0moeu

Step	Hyp	Ref	Expression
1		n0 2884	. . . 4 |- (A =/= (/) <-> E.x x e. A)
2	1	biimpi 168	. . 3 |- (A =/= (/) -> E.x x e. A)
3	2	biantrurd 796	. 2 |- (A =/= (/) -> (E*x x e. A <-> (E.x x e. A /\ E*x x e. A)))
4		eu5 1805	. 2 |- (E!x x e. A <-> (E.x x e. A /\ E*x x e. A))
5	3, 4	syl6bbr 597	1 |- (A =/= (/) -> (E*x x e. A <-> E!x x e. A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   e. wcel 1300  E.wex 1326  E!weu 1771  E*wmo 1772   =/= wne 2017  (/)c0 2875

This theorem is referenced by:  holimf2 10327  nolimf2 15032

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-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-nul 2876

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