Theorem elsnc2 2441

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 2440	. 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 958   e. wcel 960  Vcvv 1814  {csn 2413

This theorem is referenced by:  el1o 4152  elnn0 6103  sn0top 7644  metelcls 7962  ringsn 8159  elch0 9121  atoml2 10305

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417

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