Theorem elsni 1827

Description: There is only one element in a singleton.

Assertion

Ref	Expression
elsni	|- (A e. {B} -> A = B)

Proof of Theorem elsni

Step	Hyp	Ref	Expression
1		elsncg 1825	. 2 |- (A e. {B} -> (A e. {B} <-> A = B))
2	1	ibi 449	1 |- (A e. {B} -> A = B)

Syntax hints:   -> wi 2   = wceq 1091   e. wcel 1092  {csn 1808

This theorem is referenced by:  elsnc2g 1831  disjsn2 1837  sssn 1852  supsn 2168  elsuci 2289  fvconst 2899  tfrlem10 2958

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812

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