Theorem elsnc 2431

Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15.

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 2430	. 2 |- (A e. V -> (A e. {B} <-> A = B))
3	1, 2	ax-mp 7	1 |- (A e. {B} <-> A = B)

Syntax hints:   <-> wb 146   = wceq 956   e. wcel 958  Vcvv 1811  {csn 2409

This theorem is referenced by:  eltp 2439  sneqr 2477  opth1 2786  opthwiener 2807  snsn0non 3125  opthprc 3221  dmsn0 3324  dmsnsn0 3325  dmsnop 3328  cnvsn 3449  funsn 3543  funconstss 3808  fsn 3834  1st2val 4095  2nd2val 4096  opelreal 5249  ltxrt 5495  sn0top 7647  hsn0elch 9120  h1de2ctlem 9478  atoml 10309  oefil2 10567

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413

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