Theorem n0moeu 3937

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 3931	. . . 4 |- ( A =/= (/) <-> E. x x e. A )
2	1	biimpi 206	. . 3 |- ( A =/= (/) -> E. x x e. A )
3	2	biantrurd 529	. 2 |- ( A =/= (/) -> ( E* x x e. A <-> ( E. x x e. A /\ E* x x e. A ) ) )
4		eu5 2496	. 2 |- ( E! x x e. A <-> ( E. x x e. A /\ E* x x e. A ) )
5	3, 4	syl6bbr 278	1 |- ( A =/= (/) -> ( E* x x e. A <-> E! x x e. A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   E!weu 2470   E*wmo 2471   =/= wne 2794   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  minveclem4a  23201