Theorem elsnc2 2437

Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set.

Hypothesis

Ref	Expression
elsnc2.1	|- B e. V

Assertion

Ref	Expression
elsnc2	|- (A e. {B} <-> A = B)

Proof of Theorem elsnc2

Step	Hyp	Ref	Expression
1		elsnc2.1	. 2 |- B e. V
2		elsnc2g 2436	. 2 |- (B 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:  el1o 4146  elnn0 6101  sn0top 7647  metelcls 7965  ringsn 8163  elch0 9126  atoml2 10310

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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