Theorem elsnc2 2433

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 2432	. 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 954   e. wcel 956  Vcvv 1807  {csn 2405

This theorem is referenced by:  el1o 4136  elnn0 6056  sn0top 7597  metelcls 7916  ringsn 8115  elch0 9065  atoml2 10247

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-un 2046  df-sn 2408  df-pr 2409

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