Theorem elsnc2g 2436

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.

Assertion

Ref	Expression
elsnc2g	|- (B e. C -> (A e. {B} <-> A = B))

Proof of Theorem elsnc2g

Step	Hyp	Ref	Expression
1		elsni 2432	. 2 |- (A e. {B} -> A = B)
2		eleq1 1534	. . 3 |- (A = B -> (A e. {B} <-> B e. {B}))
3		snidg 2433	. . 3 |- (B e. C -> B e. {B})
4	2, 3	syl5cbir 211	. 2 |- (B e. C -> (A = B -> A e. {B}))
5	1, 4	impbid2 518	1 |- (B e. C -> (A e. {B} <-> A = B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  {csn 2409

This theorem is referenced by:  elsnc2 2437  elsuc2g 3037  efif1lem5 8734

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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