Theorem elsnc2g 2440

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 2436	. 2 |- (A e. {B} -> A = B)
2		eleq1 1537	. . 3 |- (A = B -> (A e. {B} <-> B e. {B}))
3		snidg 2437	. . 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 520	1 |- (B e. C -> (A e. {B} <-> A = B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  {csn 2413

This theorem is referenced by:  elsnc2 2441  elsuc2g 3043  efif1lem5 8729

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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