Theorem elsnc 3756

Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)

Hypothesis

Ref	Expression
elsnc.1	|- A e. _V

Assertion

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

Proof of Theorem elsnc

Step	Hyp	Ref	Expression
1		elsnc.1	. 2 |- A e. _V
2		elsncg 3755	. 2 |- (A e. _V -> (A e. {B} <-> A = B))
3	1, 2	ax-mp 5	1 |- (A e. {B} <-> A = B)

Syntax hints:   <-> wb 176   = wceq 1642   e. wcel 1710  _Vcvv 2859  {csn 3737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sn 3741

This theorem is referenced by:  sneqr  3872  sniota  4369  nnsucelrlem1  4424  nnsucelrlem3  4426  nnsucelr  4428  nndisjeq  4429  ltfinex  4464  ltfintrilem1  4465  ssfin  4470  eqtfinrelk  4486  oddfinex  4504  evenoddnnnul  4514  evenodddisjlem1  4515  nnadjoinlem1  4519  vfinspss  4551  dfop2lem1  4573  setconslem2  4732  dmsnn0  5064  dmsnopg  5066  cnvsn  5073  rnsnop  5075  funsn  5147  iunfopab  5204  funconstss  5406  fsn  5432  fvclss  5462  1p1e2c  6155  fce  6188  dmfrec  6316